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Solving the Neutron Diffusion Equation with a Radioactive Source Term
Published in Robert E. Masterson, Introduction to Nuclear Reactor Physics, 2017
Finally, anyone who has worked with differential equations knows that there are many similarities between the neutron diffusion equation and other differential equations of classical physics. Before delving into the topic of time-dependent reactor behavior, we would first like to take a moment to review some of these similarities. Suppose that we would like to find a solution to the neutron diffusion equation in a region of empty space (i.e., in a region where a vacuum exists). In this case, the value of the material buckling B2 is zero and in empty space, the value of the diffusion coefficient D is infinite. Then the steady state diffusion equation becomesand the astute reader may recognize as Laplace’s equation. Next, suppose that we would like to apply the diffusion equation to a pure scattering material where there is no fission or absorption, but where a significant amount of scattering may occur. In this case, the neutron diffusion equation becomeswhich the astute reader may recognize as Poisson’s equation. Solutions to Poisson’s equation are different than the solutions to Laplace’s equation, and some of these differences were highlighted in Chapter 13. Finally, consider the steady-state neutron diffusion equation in its conventional form. It turns out that this version of the diffusion equation is equivalent to another famous second-order differential equation called the Helmholtz equation, which was analyzed in great detail by Hermann von Helmholtz (see Figure 14.26) over 100 years ago.The Schrödinger Wave Equation, which we briefly mentioned in Chapter 1 of our companion book*, is one of the best-known examples of the Helmholtz equation. However, if one looks more closely into the subject there are many others. For example, the Helmholtz equation can also be used to predict the flow of heat through a nuclear fuel rod.† This thermal heat conduction process is discussed in reactor thermal hydraulic books and it is very important to the field of nuclear science and engineering as a whole.
FEM-BEM analysis of tyre-pavement noise on porous asphalt surfaces with different textures
Published in International Journal of Pavement Engineering, 2019
The mathematical formulations and practical applications of BEM have been discussed in detail in the previous publications (Shaw 1979, Brebbia et al.1984, Parlett 1989). The governing partial differential equation for linear acoustics in the frequency domain is the Helmholtz equation. The integral form of Helmholtz equation on boundary connects the boundary pressure p with normal velocity vn by a matrix equation as shown in Equation (3).