Fluid management calculation

In a previous paper [6], we proposed a method for modeling the availability of middle/low pressure gas networks, which consists in an interleaved execution of i) a quasi-static fluid-dynamic analysis of the network and ) a stochastic model of the failure management procedure. The latter uses non-Markovian temporal parameters, thus overcoming the limits of memoryless and imbounded support of exponential distributions. As a distinctive trait, fluid-dynamic calculations are decoupled from the non-Markovian stochastic analysis and the complexity of stochastic analysis is insensitive to topology and size of the gas network. 

For dense gas-solid two-phase flows, a four-way coupling is required however, the coupling between particles is managed in a natural way in DPMs. The task is, therefore, only to find a two-way coupling between the gas and the solid phases, which satisfies Newton s third law. Basically, the gas phase exerts two forces on particle a a drag force Vda due the fluid-solid friction at the surface of the spheres, and a force Vpa = -Va Vp due to the pressure gradient Vp in the gas phase. We will next describe these forces in more detail, along with the procedure to calculate void fraction, which is an essential quantity in the equations for the gas-solid interaction. 

The S02 oxidation process is kinetically limited, inducing a non-negligible anodic overvoltage, given by a Tafel electrokinetic law [11], In order to account for the secondary potential distribution prevailing in the domain, potential jumps are calculated at the anode/fluid interface thanks to the interfacial-type elements provided within the solver. These zero-width finite elements allow electrical potential discontinuities to be managed, as described in ref. [7], The same formalism is used at the cathode/fluid interface as well to model the proton reduction overvoltage. 

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