We model equilibrium allocations in a distribution network as the solution of a linear program (LP) which minimizes the cost of unserved demands across nodes in the network. The constraints in the LP dictate that once a given node's supply is exhausted, its unserved demand is distributed among neighboring nodes. All nodes do the same and the resulting solution is the equilibrium allocation. Assuming that the demands are random (following a jointly Gaussian law), our goal is to study the probability that the optimal cost (i.e. the cost of unserved demands in equilibrium) exceeds a large threshold, which is a rare event. Our contribution is the development of importance sampling and conditional Monte Carlo algorithms for estimating this probability. We establish the asymptotic efficiency of our algorithms and also present numerical results which illustrate strong performance of our procedures.
Water hammer phenomenon is a significant pressure wave in pipe system caused by momentum change when the moving fluid is forced to stop or change direction instantaneously. Common causes of water hammer are sudden valve closing at the end of a pipeline system, pump failure, check valve slam etc. The steam transportation pipeline system may also be vulnerable to water hammer when it confronts with the situation where liquid and steam co-exist. Water hammer often occurs when steam condenses into water in a horizontal section of steam piping. Then steam “picks up” water to form a high-velocity “slug” and create extra stress to pipe. When steam is trapped into sub-cooled water, the collapse of vapor cavity can lead to collision of two columns of liquid, resulting in a large rise in pressure which will damage pipes, supporting structures and hydraulic machinery. Nuclear power plant is composed of complex equipments and piping systems, lots of which contain both liquid and steam. Hence, there is a potential threat of occurrence of water hammer to the normal operation of systems. Thus, this phenomenon needs to be well investigated and prevented with some effective methods. For the purpose of overpressure relief under severe accidents, the spent fuel pool cooling system of CAP1000 series nuclear power plant provides a discharge passage from containment to spent fuel pool. When the containment pressure exceeds the control value, valve is opened to discharge high-temperature and high-pressure steam until the pressure drops to a safety value. During this process, serious water hammer happens, causing pressure rise beyond the design pressure and further leading to damages to pipes and structures. Therefore, water hammer of overpressure discharge pipeline in CAP1000 plant is studied in this work. On the basis of verification of the capabilities of computational code RELAP5/MOD3.3, hydraulic transient of water hammer is simulated under different conditions. It is indicated that after steam discharge stops, residual steam in pipe condenses because of contact with sub-cooled water in spent fuel pool. Subsequently, the rapid backflow and vapor cavity lead to a severe water hammer. The detailed analysis has shown that water temperature of spent fuel pool has a decisive influence on the mechanism of water hammer phenomenon, including collision of liquid column to valve disc and cavity collapse in the horizontal pipe. The collision and separation of liquid column result in relatively lower pressure amplitude.
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