An analytical formulation to describe the vibration of a fluid-shell system representing a fast reactor

Modern pool-type fast reactors comprise of containers with fluid inside. Besides holding the fluids, the containers are also required to transfer heat from one side to another smoothly. A “thin” container serves the purpose of heat transfer well. However, such containers and associated support systems may be vulnerable during strong earthquake shaking. A possible mitigating strategy is to “isolate” the reactor from horizontal earthquake shaking. Natural period of vibration for a system with horizontal seismic isolation may vary in the range of 1 – 4 seconds, and that corresponding to sloshing of fluids inside the containers may range between 3 – 8 seconds. Possibility of amplification in the sloshing response due to seismic isolation needs to be studied. Such a study can be performed through experiments, or verified and validated analytical models. A third way could be to consider rather simple systems representing a pool-type fast reactor and develop closed-form analytical solutions to describe the vibration of these systems. Such solutions would allow understanding the influence of seismic isolation on the sloshing response for a wide range of geometric and material properties of the simplistic reactor and seismic isolation systems. This paper presents the development of a closed-form analytical solution for a representative pool-type fast reactor. The system comprises of two cylindrical shells and a centrally-placed internal body. The outermost cylindrical shell is supported at the top (through open end). The inner cylinder is supported at the base of the outermost cylinder. These two cylinders are referred to as main vessel and inner vessel, respectively. Fluid is present in the inner vessel and in the annulus between the inner and main vessels. The closed end of the innermost cylinder is at the top, and is attached to the ground. This cylinder is hollow and is partially submerged in the fluid inside the inner vessel. Two broad sets of modes of vibration are observed in such systems: 1) sloshing, and 2) bulging. Analytical formulation presented in this paper considers both. Shells are assumed elastic and their behavior is characterized using Flugee’s shell theory. The behavior of the fluid is described by a velocity potential function, which considers coupling between bulging and/or sloshing modes. Natural frequencies and mode shapes of the system are calculated by minimizing the Rayleigh-Ritz quotient. The analytical modeling approach is validated against experiments conducted in the past. A parametric study is carried out to understand the effect of the ratio of radii of main and inner vessels, and density of the fluid on the dynamic properties of the system. Influence of considering coupling between the bulging and sloshing modes on the natural frequencies of the system supported by ground (non-isolated) was found small. Therefore, bulging and sloshing modes can be treated independently for a non-isolated system. Bulging frequency of the system decreases with an increase in the density of fluid or a reduction in the ratio of the radii of main and inner vessels. First mode sloshing frequencies are not considerably affected by the two factors. Second mode sloshing frequency decreases considerably with an increase in the ratio of the radii of main and inner vessels.