Graduate Thesis Or Dissertation
 

Describing function theory as applied to thermal and neutronic problems

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  • Describing functions have traditionally been used to obtain the solutions of systems of ordinary differential equations. In this work the describing function concept has been extended to include nonlinear, distributed parameter partial differential equations. A three-stage solution algorithm is presented which can be applied to any nonlinear partial differential equation. Two generalized integral transforms were developed as the T-transform for the time domain and the B-transform for the spatial domain. As specific applications of the solution technique two cases are considered: the heat conduction equation and the neutron diffusion equation. The thermal diffusion describing function (TDDF) is developed for conduction of heat in solids and a general iterative solution along with convergence criteria is presented. The proposed solution method is used to solve the problem of heat transfer in nuclear fuel rods with annular fuel pellets. As a special instance the solid cylindrical fuel pellet is examined. A computer program is written which uses the describing function concept for computing fuel pin temperatures in the radial direction during reactor transients. It was found that the quasi-linear method used in the describing function method is as accurate as nonlinear treatments and as fast as true linearization methods. The second problem investigated was the neutron diffusion equation which is intrinsically different from the first case. Although, for most situations, it can be treated as a linear differential equation, the describing function method is still applicable. A describing function solution is derived for two possible cases: constant diffusion coefficient and variable diffusion coefficient. Two classes of describing functions are defined for each case which portray the leakage and absorption phenomena. A study of the convergence criteria is also included. For the specific case of a slab reactor criticality problem the comparison between analytical and describing function solutions revealed an excellent agreement.
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