Four-fluid Model Research Articles

Development of a four-fluid model for condensing steam flows

A new four-fluid model is proposed for condensing water-steam flows applicable to a wide variety of different flow conditions relevant for nuclear power plants. The model expands upon current state-of-the-art three-fluid models by adding a second droplet fluid phase to account for homogeneous nucleation of droplets from the bulk vapor field. Consistent with prior three-fluid models, liquid wall film and entrained droplet fluids share a common temperature and energy equation, resulting in a system of eleven conservation equations. An additional governing PDE is required for the nucleated droplet number density, for twelve total governing equations. A full set of closure relations for the interphase and wall interactions is given. An initial test of the model formulation in a quasi-one-dimsensional converging-diverging nozzle with condensing steam was carried out using a simple in-house solver as a proof of concept. Results showed qualitatively correct trends compared with experimental data.

Modeling and assessment of two-phase transonic steam flow with condensation through the convergent-divergent nozzle

In the Fukushima accident, it was found that the Reactor Core Isolation Cooling (RCIC) system played a crucial role in delaying core meltdown by almost three days in Fukushima Daiichi Unit 2, because of self-regulated operation of the steam driven RCIC turbine-pump injection system. Steam flow in the convergent-divergent nozzles of the RCIC Terry turbine is two-phase non-equilibrium transonic flow with homogenous nucleation condensation. To more accurately predict the dynamic process and behavior of the transonic compressible steam flow, a one-dimensional transient two-phase analytical model is presented. A simplified four-fluid model was employed in the present work with the consideration of four separate fluid fields: vapor, liquid film, entrained droplets and condensed droplets. The mass, momentum and energy interactions between the fluids were considered and modeled. An extended seven-equation non-equilibrium critical flow model was developed to obtain the critical pressure and velocities of each phase at the nozzle throat. To predict the wetness in the divergent section, a mechanistic nucleation condensation model was integrated in the nozzle analysis model, considering the generation and consequent growth of droplets. The governing differential equations on a staggered grid were discretized using the second-order Lax-Wendroff scheme with a flux limiter, and the Semi-Implicit Method for Pressure-Linked Equation (SIMPLE) algorithm was employed to solve the discrete linear system. To demonstrate the predictability and reliability of the physical models and the numerical method proposed in the present work, three representative nozzles were modeled and simulated. The results show good agreement with the available experimental data, even for condensation shock.

Stability analysis of a periodic system of relativistic current filaments

The nonlinear evolution of current filaments generated by the Weibel-type filamentation instability is a topic of prime interest in space and laboratory plasma physics. In this paper, we investigate the stability of a stationary periodic chain of nonlinear current filaments in counterstreaming pair plasmas. We make use of a relativistic four-fluid model and apply the Floquet theory to compute the two-dimensional unstable eigenmodes of the spatially periodic system. We examine three different cases, characterized by various levels of nonlinearity and asymmetry between the plasma streams: a weakly nonlinear symmetric system, prone to purely transverse merging modes; a strongly nonlinear symmetric system, dominated by coherent drift-kink modes whose transverse periodicity is equal to, or an integer fraction of the unperturbed filaments; a moderately nonlinear asymmetric system, subject to a mix of kink and bunching-type perturbations. The growth rates and profiles of the numerically computed eigenmodes agree with particle-in-cell simulation results. In addition, we derive an analytic criterion for the transition between dominant filament-merging and drift-kink instabilities in symmetric two-beam systems.

Simulation study of the sheath region of a processing plasma with two-temperature electrons and charged nanoparticles

A four-fluid model is used to investigate the structure of a dusty plasma sheath with two species of electrons, i.e., cold and hot electrons. Numerical results show that, in the presence of hot electrons, regular fluctuations are developed in the spatial profiles of the sheath potential and number densities of the plasma and dust nanoparticles. The amplitude and spatial period of the fluctuations depend on the hot electron parameters. The sheath width shows a non-monotonic dependence on the temperature and number density of the hot electrons, as well as the density and Mach number of the nanoparticles.

FOUR-FLUID MODEL AND NUMERICAL SIMULATIONS OF MAGNETIC STRUCTURES IN THE HELIOSHEATH

The first part of this paper extends the three-fluid model of Avinash & Zank for magnetic structures in the heliosheath to a four-fluid model consisting of electrons, pick-up ions (PUIs), solar wind ions (SWIs), and neutral hydrogen. The PUIs are generated by neutrals via charge exchange with SWI. Since the kinetic pressure of PUI is nearly three to four times the pressure of SWI, these are more suited to mediate small-scale structures in the heliosheath such as magnetic holes (MH)/humps etc. The constant energy exchange between these two fluids drives them nonadiabatic. The PUIs are isothermal (γ = 1) while SWIs are nonadiabatic with an index γ ≈ 1.25. The four-fluid model captures these effects via a modified equation of state for PUI and SWI. The phase space of time-independent solutions in terms of the Mach numbers of PUI and SWI is constructed to delineate the parameter space which allows structure formation in the heliosheath. The second part of the paper examines the stability of the time-independent solutions computed in the first part by evolving them via a full system of Hall-MHD equations. The simulation results show that these solutions are not quite stable. As the structure propagates it develops growing oscillations in the wings. Concomitantly, there are changes in the amplitude and width of the structure. This instability could be due to local changes in the velocity of the structure and reflects an exchange between the kinetic and magnetic parts of the total energy. Our results about the presence of growing oscillations in the wings of solitary wave solutions are consistent with the recent analysis of MHs in the heliosheth by Burlaga et al. Their analysis also shows evidence for the presence of oscillations and instabilities in the wings of MHs in the heliosheath.

Four-Fluid Model and Radial Transport Equation for Toroidal Momentum in Tokamak Plasmas

Mechanically, the usually assumed transport equation of the toroidal momentum in tokamak plasmas seems to be questionable since it fails to consider constraint conditions on the motion of the trapped particles. In this paper, starting from the so-called four-fluid model which can make up the above-mentioned deficiency of the ordinary MHD model and can correctly and effectively describe the relations between the particle flows and the acting forces, a more reasonable radial transport equation for the toroidal momentum is derived on the basis of the ambipolarity of the radial particle fluxes.

A Four-Fluid Model and Analysis of m = 1 Internal Kink Instability

Starting from the four-fluid equation which can simply and effectively describe the motion of trapped particles, we derive the eigenmode equations and the dispersion relation for m=1 internal kink mode with the effects of trapped particles, diamagnetic drifts, finite resistivity and anomalous electron viscosity included. Then, using those results, we find the trapped particles effects have substantial influence on m=1 internal kink mode in the cases with and without high-energy alpha particles in the core plasma.

A trapped-passing fluid model for tokamak neoclassical transport

In the low collisionality banana regime, the response of the passing and trapped particles to toroidal forces is essentially distinct: Passing particles accelerate, whereas trapped particles drift radially. In order to model current drive and particle transport in a physically intuitive manner, a four-fluid model for a simple electron–ion plasma involving separate momentum equations for each of the trapped and passing species has been formulated. Because of relative drifts, the four fluids undergo collisional friction with each other and with neutrals; a radial electric field arises with the inclusion of momentum sources, particle sources, and the collisions with neutrals. Radial plasma transport, which conserves canonical angular momentum, is modeled by scattering the configurational component of trapped particle momentum into translational momentum of the passing particles at a rate related to the density and temperature gradients by standard neoclassical theory. For given radial temperature profile, the neoclassical differential equation for the density is obtained. This equation is integrated numerically to obtain the radial density profile. Over a broad range of parameters, the volume average density is close to the empirical Hugill–Murakami–Greenwald [Nucl. Fusion 28, 2199 (1988)] density limit n̄ (1020 m−3)=J̄ (MA m−2); although most plasmas operate below this density, the results suggest that near the density limit, plasma particle confinement may be neoclassical. In this case, it is shown explicitly, in the steady state, that Ohm’s law can be expressed very simply in terms of the classical Spitzer conductivity with no trapped particle correction, a fact first noted by Ware [Nucl. Fusion 13, 793 (1973)]. Alternately, the effect of the trapped particle correction on conductivity cancels the bootstrap current. Concomitantly, deposition of rf wave momentum on thermal, trapped particles in the direction so as to drive plasma current, will pinch the particles inward and result in toroidal current independent of the trapping effect.