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.
231 citations
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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
21 citations
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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.
19 citations
Dissertation•
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01 Jan 2014
TL;DR: In this paper, the single and two-phase heat transfer in high aspect ratio mini-channels has not been well-characterized, especially in regard to the onset of nucleate boiling.
Abstract: Heat transfer in high aspect ratio mini-channels has important applications for materials test reactors using plate-type fuel. These fuel plates typically possess coolant channels with hydraulic diameters on the order of 4 mm or less. The single and two-phase heat transfer in such channels has not been well-characterized, especially in regard to the onset of nucleate boiling. While surface effects are known to dramatically influence the incipience of boiling, they have not been widely considered under forced convection. Since the limiting safety system setting for the MITR is the onset of nucleate boiling, there is considerable interest in better characterizing the phenomenon in such channels. This study presents a first-of-a-kind, two-phase flow facility designed to measure the singlephase heat transfer coefficient and onset of nucleate boiling in a high aspect ratio mini-channel over a wide range of flow conditions while also permitting high speed visualization of the entire surface. The single-phase heat transfer coefficient is measured for mass fluxes ranging from 750 kg/m2-sec up to 6000 kg/m2-sec and for subcoolings ranging from 20 °C to 70 °C. The onset of nucleate boiling superheat and heat flux are measured for mass fluxes ranging from 750 kg/m2sec to 3000 kg/m2-sec and for subcoolings ranging from 10 °C to 45 °C. Measurements are supported with high speed videography to visualize bubble incipience when conditions permit. The influence of surface wettability on the incipience point is also investigated by performing tests on a surface oxidized at high temperature in air. Using a boundary layer analysis along with experimental data obtained in the study, a semianalytical correlation is developed to predict the single-phase heat transfer coefficient in high aspect ratio rectangular channels. The correlation accounts for effects from secondary flows and heating asymmetry, and is suitable for both the transition and fully turbulent flow regimes. The new correlation predicts the Nusselt number with a mean absolute error of 4.9% in the range of 2.2
17 citations
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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.
14 citations
References
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TL;DR: In this paper, the correlation exponent v is introduced as a characteristic measure of strange attractors which allows one to distinguish between deterministic chaos and random noise, and algorithms for extracting v from the time series of a single variable are proposed.
Abstract: We study the correlation exponent v introduced recently as a characteristic measure of strange attractors which allows one to distinguish between deterministic chaos and random noise. The exponent v is closely related to the fractal dimension and the information dimension, but its computation is considerably easier. Its usefulness in characterizing experimental data which stem from very high dimensional systems is stressed. Algorithms for extracting v from the time series of a single variable are proposed. The relations between the various measures of strange attractors and between them and the Lyapunov exponents are discussed. It is shown that the conjecture of Kaplan and Yorke for the dimension gives an upper bound for v. Various examples of finite and infinite dimensional systems are treated, both numerically and analytically.
5,023 citations
"Non-linear dynamics of a two phase ..." refers methods in this paper
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TL;DR: Finite Difference Methods in Heat Transfer as mentioned in this paper presents a step-by-step delineation of finite difference methods for solving engineering problems governed by ordinary and partial differential equations, with emphasis on heat transfer applications.
Abstract: Finite Difference Methods in Heat Transfer presents a clear, step-by-step delineation of finite difference methods for solving engineering problems governed by ordinary and partial differential equations, with emphasis on heat transfer applications The finite difference techniques presented apply to the numerical solution of problems governed by similar differential equations encountered in many other fields Fundamental concepts are introduced in an easy-to-follow mannerRepresentative examples illustrate the application of a variety of powerful and widely used finite difference techniques The physical situations considered include the steady state and transient heat conduction, phase-change involving melting and solidification, steady and transient forced convection inside ducts, free convection over a flat plate, hyperbolic heat conduction, nonlinear diffusion, numerical grid generation techniques, and hybrid numerical-analytic solutions
593 citations
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TL;DR: In this article, the first minimum of the logarithm of the generalized correlation integral C1(τ) provides an easily evaluable criterion for the proper choice of the time delay τ that is needed to reconstruct the trajectory in phase space from chaotic scalar time series data.
Abstract: It is shown that the first minimum of the logarithm of the generalized correlation integral C1(τ) provides an easily evaluable criterion for the proper choice of the time delay τ that is needed to reconstruct the trajectory in phase space from chaotic scalar time series data.
298 citations
"Non-linear dynamics of a two phase ..." refers methods in this paper
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TL;DR: In this article, an experimental study on the onset of thermally induced two-phase flow oscillations has been carried out in a uniformly heated boiling channel using Freon-113 as the operating fluid.
Abstract: An experimental study on the onset of thermally induced two-phase flow oscillations has been carried out in a uniformly heated boiling channel using Freon-113 as the operating fluid. The effects of inlet subcooling, system pressure, inlet and exit restrictions, and inlet velocity have been studied. The experimental data have been compared with the equilibrium as well as the nonequilibrium theory including the effect of subcooled boiling. It has been found that the effect of thermal nonequilibrium should be included in a theoretical model for accurate prediction of the onset and the frequency of thermally induced flow oscillations. A simplified stability criterion has also been presented and compared with the experimental data.
147 citations
"Non-linear dynamics of a two phase ..." refers background in this paper
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