Non-linear dynamics of a two phase flow system in an evaporator: The effects of (i) a time varying pressure drop (ii) an axially varying heat flux
TL;DR: In this paper, the authors study the phenomena of density wave oscillations (DWO) in a vertical heated channel and use the homogeneous equilibrium model to simulate the flow in the two-phase region.
Abstract: In this paper we study the phenomena of density wave oscillations (DWO) in a vertical heated channel. The homogeneous equilibrium model is used to simulate the flow in the two-phase region. The equations are solved numerically using a `shooting-method' technique. This in its turn employs an implicit backward finite difference scheme. The scheme can incorporate the movement of the interface. It is very elegant and does not involve storage of variables in large N×N matrices. This scheme is sufficiently general and can be used to simulate the dynamic behaviour when: (i) the heat flux imposed at the surface is non-constant, i.e. exhibits an axial variation; and (ii) the imposed pressure drop is varied periodically at a fixed frequency. A possible explanation for the conflicting reports of the effect of a periodic variation in heat flux is provided using a linear stability analysis and the D-partition method. The interaction of the natural frequency of the DWO and the fixed forcing frequency of the imposed pressure drop gives rise to various phenomena viz relaxation oscillations, sub-harmonic oscillations, quasi-periodic and chaotic solutions. To aid the experimentalist describe this infinite-dimensional system on the basis of his experimental results we discuss the characterisation using only the velocity time series data. This is done employing the method of delay coordinate embedding. The phase portraits, stroboscopic map and correlation dimension of the actual attractor are compared with that of the reconstructed attractor from the velocity time series.
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TL;DR: An updated review of two-phase flow instabilities including experimental and analytical results regarding density-wave and pressure-drop oscillations, as well as Ledinegg excursions, is presented in this article.
Abstract: An updated review of two-phase flow instabilities including experimental and analytical results regarding density-wave and pressure-drop oscillations, as well as Ledinegg excursions, is presented. The latest findings about the main mechanisms involved in the occurrence of these phenomena are introduced. This work complements previous reviews, putting all two-phase flow instabilities in the same context and updating the information including coherently the data accumulated in recent years. The review is concluded with a discussion of the current research state and recommendations for future works.
292 citations
TL;DR: In this article, a transient thermal hydraulic model is developed with a characteristics-based implicit finite-difference scheme to solve the nonlinear mass, momentum and energy conservation equations in a time-domain.
Abstract: The objective of the paper is to analyze the thermally induced density wave oscillations in water cooled boiling water reactors A transient thermal hydraulic model is developed with a characteristics-based implicit finite-difference scheme to solve the nonlinear mass, momentum and energy conservation equations in a time-domain A two-phase flow was simulated with a one-dimensional homogeneous equilibrium model The model treats the boundary conditions naturally and takes into account the compressibility effect of the two-phase flow The axial variation of the heat flux profile can also be handled with the model Unlike the method of characteristics analysis, the present numerical model is computationally inexpensive in terms of time and works in a Eulerian coordinate system without the loss of accuracy The model was validated against available benchmarks The model was extended for the purpose of studying the flow-induced density wave oscillations in forced circulation and natural circulation boiling water reactors Various parametric studies were undertaken to evaluate the model's performance under different operating conditions Marginal stability boundaries were drawn for type-I and type-II instabilities in a dimensionless parameter space The significance of adiabatic riser sections in different boiling reactors was analyzed in detail The effect of the axial heat flux profile was also investigated for different boiling reactors
26 citations
TL;DR: In this paper, the authors analyze DWO in several boiling channels with varying lengths (Froude number) adopting moving node scheme and fixed node scheme (FNS) to evaluate the capability of the methods.
Abstract: The research on density wave oscillation (DWO) in boiling channels during the last few decades has been reviewed. Model reductions through lumped parameterization of the distributed channels have been exercised to compute nonlinear DWOs. In the present article, we attempt to analyze DWOs in several boiling channels with varying lengths (Froude number) adopting moving node scheme (MNS) and fixed node scheme (FNS). Relative performances of MNS and FNS have been analyzed to evaluate the capability of the methods. The analysis suggests that MNS is highly computationally efficient and has excellent convergence compared to FNS and finite difference method. Extended numerical oscillations have been observed in FNS. The analysis also suggests that DWOs are strongly coupled with the extent of inlet subcooling (boiling boundary), pressure drop and vapor quality. At high inlet subcooling, the ratio of time period to transit time is significantly higher than 2.0 (2.5–6.0) whereas at low inlet subcooling the ratio remains around 2.0.
Numerical experiments on long boiling channels (low Froude number) and short ones (high Froude number) derives a clear difference that the short channels with high Froude number has “islands of instability” in Npch–Nsub plane and undergoes both supercritical and subcritical bifurcations, whereas the boiling channel with low Froude number undergoes only supercritical bifurcations. The effect of node numbers on marginal stability boundary (MSB) has been discussed. Increased speed of convergence is observed with higher number of nodes. With finer nodalizations, the region of instability extends. Extensive validations of the nonlinear models with reference experimental data and numerical results confirm that MNS satisfactorily predicts MSB, supercritical and subcritical bifurcations. Quasi-periodic en route to chaos has been detected in the boiling channel as a result of periodic perturbation of pressure drop (Eu). The same has been confirmed by the analysis of power spectrum density (PSD) and computation of Lyapunov exponents.
22 citations
TL;DR: In this article, a nuclear coupled thermal-hydraulic model was developed to simulate core-wide and regional stability analysis in time domain within the limitation of desktop research facility for a boiling water reactor subjected to operational transients.
Abstract: The objective of the paper is to develop a nuclear coupled thermal-hydraulic model in order to simulate core-wide (in-phase) and regional (out-of-phase) stability analysis in time domain within the limitation of desktop research facility for a boiling water reactor subjected to operational transients. The integrated numerical tool, which is a combination of thermal-hydraulic, neutronic and fuel heat conduction models, is used to analyze a complete boiling water reactor core taking into account the strong nonlinear coupling between the core neutron dynamics and primary circuit thermal-hydraulics via the void-temperature reactivity feedback effects. The integrated model is validated against standard benchmark and published results. Finally, the model is used for various parametric studies and a number of numerical simulations are carried out to investigate core-wide and regional instabilities of the boiling water reactor core with and without the neutronic feedback effects. Results show that the inclusion of neutronic feedback effects has an adverse effect on boiling water reactor core by augmenting the instability at lower power for same inlet subcooling during core-wide mode of oscillations, whereas the instability is being suppressed during regional mode of oscillations in presence of the neutronic feedback. Dominance of core-wide instability over regional mode of oscillations is established for the present case of simulations which indicates that the preclusion of the former will automatically prevent the latter at the existing working condition.
19 citations
TL;DR: In this article, the period of density wave oscillations in an uniformly heated horizontal test section is experimentally investigated for a 5mm I.D. pipe where R134a is used as working fluid.
Abstract: The period of Density Wave Oscillations (DWOs) in an uniformly heated horizontal test section is experimentally investigated. The test section consists of a 5 mm I.D. pipe where R134a is used as working fluid. The experiment is performed for a range of inlet pressures P i [500–1200 kPa], inlet sub-cooling [10 and 20 K], maintaining constant heat fluxes q ″ [38 kW/m 2 ] and mass flux G [300 kg/m 2 s]. The effect of the system parameters on the period of the DWOs is studied. It is observed that the period of the DWOs increases as inlet pressure and inlet sub-cooling temperature increase. Furthermore it is observed changes in the period which might be connected to the changes in the flow regime distribution inside the pipe.
19 citations
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TL;DR: In this article, the drift flux model is used to make the set of equations dimensionless to ensure the mutual independence of the dimensionless variables and parameters: the steady-state inlet velocity v, the inlet subcooling number N sub and the phase change number N pch.
Abstract: Linear and nonlinear mathematical stability analyses of parallel channel density wave oscillations are reported. The two phase flow is represented by the drift flux model. A constant characteristic velocity v 0 ∗ is used to make the set of equations dimensionless to ensure the mutual independence of the dimensionless variables and parameters: the steady-state inlet velocity v , the inlet subcooling number N sub and the phase change number N pch . The exact equation for the total channel pressure drop is perturbed about the steady-state for the linear and nonlinear analyses. The surface defining the marginal stability boundary (MSB) is determined in the three-dimensional equilibrium-solution/operating-parameter space v − N sub − N pch . The effects of the void distribution parameter C 0 and the drift velocity V g j on the MSB are examined. The MSB is shown to be sensitive to the value of C 0 and comparison with experimental data shows that the drift flux model with C 0 > 1 predicts the experimental MSB and the neighboring region of stable oscillations (limit cycles) considerably better than do the homogeneous equilibrium model ( C 0 = 1, V g j = 0 ) or a slip flow model. The nonlinear analysis shows that supercritical Hopf bifurcation occurs for the regions of parameter space studied; hence stable oscillatory solutions exist in the linearly unstable region in the vicinity of the MSB. That is, the stable fixed point v becomes unstable and bifurcates to a stable limit cycle as the MSB is crossed by varying N sub and/or N pch .
97 citations
TL;DR: In this article, a quasilinear Hopf-bifurcation analysis of the marginal stability boundary of a uniformly heated boiling channel is presented. But the analysis is restricted to the case when the effects of gravity and friction are considered.
Abstract: Thermally induced flow instabilities in uniformly heated boiling channels have been studied analytically. The classical homogeneous equilibrium model was used. This distributed model was transformed into an integrodifferential equation for inlet velocity. A linear analysis showed interesting features (i.e. islands of instability) of the marginal stability boundary which appear when the effects of gravity and friction were systematically considered. A quasilinear Hopf-bifurcation analysis, valid near the marginal-stability boundaries, gives the amplitude and frequency of limit-cycle oscillations that can appear on the unstable side of the boundary. The analysis also shows cases where a finite-amplitude perturbation can cause a divergent instability on the stable side of the linear-stability boundary.
95 citations
"Non-linear dynamics of a two phase ..." refers methods in this paper
...The method of characteristics employed by Achard et al. (1985) can be used only when the heat flux is constant....
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...The first detailed theoretical study of DWO was carried out by Achard et al. (1985) using the homogeneous equilibrium model (HEM) for the two phase flow....
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...The linear stability analysis and the determination of the stability boundaries using the D-partition method are discussed in Achard et al. (1985) for the case of a constant Npch....
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...The first detailed theoretical study of DWO was carried out by Achard et al. (1985) using the homogeneous equilibrium model (HEM) for the two phase flow. They obtained the marginal stability boundary (MSB) across which DWO occur and the boundary across which Ledinegg instability occurs (Ledinegg stability boundary or LSB) in the two-dimensional parameter space of subcooling number and friction number using the D-partition method. They transformed their system of hyperbolic equations to an integro-differential system in inlet velocity using the method of characteristics for their analysis. Rizwan-uddin and Dorning (1986) followed the method of Achard et al. and used the drift-flux model. They obtained the MSB for DWO in the parameter space of sub-cooling number and phase change number and compared their results with experiments. In a later study (Rizwan-uddin and Dorning, 1988, 1990) the effect of a periodic perturbation in the imposed pressure drop on the system behaviour was considered. It was shown that the system can exhibit sub-harmonic or quasi-periodic or chaotic behaviour depending on the amplitude and the frequency of the periodic forcing. A limitation of this approach based on the method of characteristics is that it can be applied only when the heat flux at the surface is independent of axial position. The numerical codes existing in the literature like PHOENICS, SETS assume a pressure variation in the channel (Jones (1987))....
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...The first detailed theoretical study of DWO was carried out by Achard et al. (1985) using the homogeneous equilibrium model (HEM) for the two phase flow. They obtained the marginal stability boundary (MSB) across which DWO occur and the boundary across which Ledinegg instability occurs (Ledinegg stability boundary or LSB) in the two-dimensional parameter space of subcooling number and friction number using the D-partition method. They transformed their system of hyperbolic equations to an integro-differential system in inlet velocity using the method of characteristics for their analysis. Rizwan-uddin and Dorning (1986) followed the method of Achard et al....
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90 citations
"Non-linear dynamics of a two phase ..." refers background in this paper
...Yadigaroglu and Bergles (1972) have reported a destabilising effect of a periodic variation of heat flux....
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TL;DR: In this article, the effect of thermal non-equilibrium between the phases has been included by deriving a constitutive equation for the mass rate of vapor generation from steady state energy consideration.
Abstract: The problem of thermally induced flow instabilities in a uniformly heated boiling channel has been studied analytically. The effect of thermal non-equilibrium between the phases has been included by deriving a constitutive equation for the mass rate of vapor generation from steady state energy consideration. The system characteristic equation is derived by introducing a small perturbation in the inlet velocity, and the system stability boundary is determined by using the D-partition method. When compared with the equilibrium theory, the present non-equilibrium theory predicts a more stable system at low subcooling number and a more unstable system at high subcooling number. When compared with experiment, the present non-equilibrium theory agrees well with the data on the system stability boundary at low subcooling and the frequency of oscillations. However, further studies are required for better prediction of the system stability at high subcooling.
85 citations
"Non-linear dynamics of a two phase ..." refers background in this paper
...Assumptions 3 and 4 have been shown to be appropriate for theoretical predictions by Saha and Zuber (1978)....
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...Assumptions 3 and 4 have been shown to be appropriate for theoretical predictions by Saha and Zuber (1978). These authors have considered a drift flux model along with thermal non-equilibrium and have found the stability boundaries quite close to those predicted by the homogeneous equilibrium model....
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TL;DR: Motivated by the enhancement of heat transfer under oscillating flow conditions in single-phase heated channels and by stability problems in two-phase systems such as those in boiling water reactor, the work in this paper was proposed.
Abstract: Motivated by the enhancement of heat transfer under oscillating flow conditions in single-phase heated channels and by stability problems in two-phase systems such as those in boiling water reactor...
53 citations