(32h) Manifestation of Acceleration During Transient Diffusion in Nuclear Fuels Processes
AIChE Spring Meeting and Global Congress on Process Safety
Monday, March 22, 2010 - 4:30pm to 4:50pm
Nuclear fuel materials are those that are capable of undergoing fission reactions along with liberation of enormous quantity of energy. Examples are Uranium, Thorium and Plutonium, etc. The nuclear fuel cycle comprises of mining the ore, processing of ore, fuel use, spent nuclear fuel processing and recycling. Per IAEA, International Atomic Energy Agency, there are 439 nuclear power plant reactors world over operational in 31 different countries. The country with the most nuclear energy is United States and in France nuclear power plants forms the highest percentage of electrical energy at 78% in 2006. Nuclear fission reactions are autocatalytic. The reaction products catalyze the reaction. An upper shape limit of a nuclear fuel rod is derived. In a nuclear pile the rate of fission depends on the local neutron concentration. It neutrons are produced at a rate that exceeds the rate of escape by diffusion the reaction is self-sustaining and a nuclear explosion occurs. In autocatalytic systems, if the ratio of the system surface to the system volume is large, then the reaction products tend to escape from the boundaries of the system. If the surface to volume ratio is small however, the rate of escape may be less than the rate of production and the reaction rate will increase rapidly. For a system of a given shape, there will be a critical size for which the rate of production just equals the rate of removal. For a long cylindrical rod, using parabolic transient Fick's laws of diffusion and simultaneous autocatalytic reaction, a critical radius for the system can be calculated as:
R(crit) = alpha*(D/k"')^.5
Where alpha is the first zero of the zero-order Bessel function J0, k1?' is the autocatalytic first order reaction rate and D binary neutron diffusivity. This limit is referred to as the shape limit of nuclear fuel rod. Above this size, autocatalytic runaway can be expected. . The generalized Fick's law of molecular diffusion and relaxation can be derived by including the acceleration term in the free electron theory, the acceleration term in the Stokes-Einstein theory for molecular diffusion, by accounting for the accumulation term in the kinetic theory of gases and combining in series the Hooke's elastic element and Newton's viscous element in the viscoelastic theory. The relaxation time was found to be a third of the collision time of the electron and the obstacle. The velocity of heat was found to be identical with the velocity of mass derived from kinetic representation of pressure or the Maxwell representation of the speed of molecules. The transient concentration profile in a long cylindrical rod during simultaneous diffusion and autocatalytic reaction using the generalized Fick's laws of molecular diffusion and relaxation. A lower limit on the radius of the fuel rod is obtained in order to avoid cycling of concentration ni the time domain during transience. The lower limit on the radius of the long cylindrical rod subject to simultaneous autocatalytic reaction and diffusion exist and found to be
R(crit) = 4.8096(Dtr)^.5/(1+ktr)
Closed form analytical solution for transient concentration profile for simulataneous autocatalytic reaction and finite speed diffusion is derived using the method of separation of variables. The solution is bifurcated. At large relaxation times, the time portion of the solution is cosinous albeit damped by the decaying exponential. Both the cycling limit and shape limits have to be considered during design of nuclear fuel rod systems. The shape limit was derived from steady state considerations. The lower cycling limit was derived from transient finite speed diffusion considerations.