Critical mass theory

With T) = 1300 °C, Too = 25 °C and 7Fj0 = 1, we have two equations in two unknowns m" and nip for the extinction conditions. However, under only suppression of the burning rate, Equation (9.80) applies, giving a nearly linear result in terms of q" and m" as well as 02,oo. The nonlinearity of the blocking effect can be ignored as a first approximation, and the experimental results can be matched to this theory. By subtracting Equation (9.81a) from Equation (9.80), we express the critical mass loss flux as... 

The self-heating and ignition of baled or loose wool in bulk storage is discussed and analysed, and steady state thermal explosion theory is applied to the prediction of critical masses and induction periods for storage and transportation situations in relation to ambient temperature. Results obtained were consistent with current safety practices. 

In the 1941 paper with Yu. B. Khariton [40], the problem of the critical size of a sample of 235 U in the fission of nuclei by fast neutrons was considered. The calculations showed that, in order to sustain a chain fission reaction by fast neutrons in a sample of 235 U surrounded by a heavy neutron reflector, it is sufficient to have only ten kilograms of pure 235U isotope. Here also a theory is given which allows calculation of the critical mass of... 

In field theory rc is known as Critical mass . Note that rc depends on cr, diverging for a — 0 (u.v. divergence). 

As expected, the distribution shifts toward low Rp values with respect to the depth, revealing a profile of crosslinking inside the elastomer. An estimate of Mc, the critical mass between knots, can be derived using De Gennes theory [34] from the measured value of Prrl , the maximum of each distribution curve. 

The value of critical mass, M., calculated in this way is, however, considerably overestimated by the elementary diffusion theory. The more exact diffusion theory, allowing for the long free path, drops R. by a factor of about 2/3 giving... 

The simplest scheme which might be autocatalytic is indicated in the sketch where the active material is disposed in a hollow shell as indicated in Fig. 8.3a. Suppose that when the firing plug is in place one has just the critical mass for this configuration. If as the reaction proceeds the expansion were to proceed only inward it is easy to see from diffusion theory that v would increase. Of course in actual fact it will proceed outward (tending to decrease v ) as well as inward and the outward expansion would in reality give the dominant effect. However, even if the outward expansion were very small compared to the inward expansion, it has been calculated that this method gives very low efficiency with 12 an efficiency of only about 10 was calculated. 

Oliver, P., Marwell, G. and Teixeira, R. Group heterogeneity, interdependence and the production of collective goods a theory of the critical mass, I. To be published in American Journal of SocicAo. ... 

Let us first consider the liquid/vapor phase transition of NaCl. PVT data have been recorded up to about 2000 K, which is still far below the critical point. Extrapolations guided by simulations and theory predict Tc = 3300 K and the critical mass density dc = 0.18 g cm-3 [33], With a = 0.276 nm and e = 1, this maps onto Tc = 0.05 and p = 0.08. 

They laid their plans, often during hikes into the uninhabited wild surroundings of the mesa. They had to rely heavily on theoretical anticipations of the effects they wanted to study that was their basic constraint. Any experimental device that demonstrated a fast-neutron chain reaction to completion would use up at least one critical mass there could be no controlled, laboratory-scale bomb tests, no squash-court demonstrations. They decided they had to analyze the explosion theoretically and work out ways to calculate the stages of its development. They needed to understand how neutrons would diffuse through the core and the tamper. They needed a theory of the explosion s hydrodynamics—the complex dynamic motions of its fluids, which the core and tamper would almost instantly become as their metals heated from solid to liquid to gas. 

A2.1.3 Besults and Dincnssion. Experiments were performed with DjO, beryllium, and H O reflectors to find (1) critical mass, (2) neutron distributions, and (3) values of control rods, cavities, etc., in terms of fuel. Various ratios of A1 to HjO were used. These experiments served as a check on the theoretical calculations of critical masses under the same conditions. The values for the critical mass found in both theory and experiment are compared in Table A2.A. 

Shear strength of a bulk material in differing states of dilation is a key property of interest for flow considerations. The conventional hopper design method for mass flow is based upon critical state theory, and a Jenike shear cell is used to secure yield locus values upon which a design procedure is based. This technique is universally accepted, but not widely used for small hoppers for various reasons. Significant cost and expertise is required to obtain accurate values, compared with full-scale trials and... 

The existence of the critical conditions at flame extinction has also been postulated by the Fire Point Theory [29] and supported by the experimental data for the critical mass pyrolysis and heat release rates [2]. 

D. S. Selengut, Critical mass calculations for hare hydrogen moderated recuitors by means of transport theory, APEX-121, September, 1952. 

W.H. Arnold, Jr., Critical Masses and Lattice Parameters of HiO-HOb Critical Experiments A Comparison of Theory and Eiqperiment, TAEC-1S2. 

The Shape Factor analyses utilized three-energy-group diffusion theory in spherical and two-dimensional r,z geometry. Table 11 is a summary of the analyses. It may be seen that the optimum geometry Shape Factor is a function of the reactor system. It is clearly not an invariant quantity. Furthermore, the variation of critical mass with core geometry also is a function of the specific reactor system. 

As a basis for comparison, material-buckling values were inferred from two-region critical mass measurements using four-group theory. Two-region critical mass measurements were carried out because only a limited number of UC-fud elements (24) were available for critical mass studies. The dependence of the uncorrected measured values iqmn size and diffusion anisotrtq can be seen in Table I, which also indudes the inferred from critical values. 

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