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
References
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DOI•
01 Jan 1969
TL;DR: In this article, the main subject of study has been the influence of subcooling, of a sine-shaped heatflux, and of a combination of both on the steady-state and dynamic characteristics of a natural-circulation, pressurised, boiling water system.
Abstract: The main subject of study has been the influence of subcooling, of a sine-shaped heatflux, and of a combination of both on the steady-state and dynamic characteristics of a natural-circulation, pressurised, boiling water system. In natural-circulation boiling systems hydrodynamic instahilities may occur at constant power. They appear to arise from a dependenee of the vapour volume production rateuponthe flow-rate as a result of energy conservation and simultaneously the flow-rate depends upon the resident vapour volume in the system as a result of momenturn conservation and continuity. The steam pressures were taken 15.5 and 30 atm, corresponding to saturation temperatures of 200°C and 234°C respectively. Although the experimental results disclose the fact that the transition from stable to unstable behaviour is not accompanied by a discontinuous change of all physical variables, preferenee has been given to a classification of the experiments in steady-state and dynamic measurements. It was preferred to incorporate the transfer fUnctionmeasurements in the dynamic part. Chapter 2 describes the experiences with void measurements by applying the impedance technique in addition to data conceming the loop dimensions and the measuring equipnent. The calibration of gauges has been referred to Maxwell' s theory. Chapter 3 deals with the measurements under steady-state conditions of the inlet velocity, the axial void distribution and pressure drops at different condit i ons of channel power, subcooling at the inlet, fluxshape and pressure. Data reduction was applied to calculate local values of the slipfactor and of two-phase friction. The slipfactor has been represented by adopting the correlations derived by Bankoff and Zuber. The values of two-phase friction were established according to ~~rtinelli-Nelson. Chapter 4 covers the results of the analysis of steady-state noise and of oscillations under unstable conditions in terms of amplitudes and 13 phase differences of ~Pinlet and the axially distributed voids. The conditions were chosen equal to those of chapter 3. The void propagation veloeities have been campared with the expression derived by Neal fram the energy equation. A limited number of transfer function measurements between the channel power as the perturbed quantity, and the dependent variables Apinlet and the various axial voids, supplemented the experiments. The void propagation velocity as a function of frequency has been compared with the expression of Zuber, basedon the theory of kinematic waves. Chapter 5 describes a theoretica! study based on a solution of the familiar laws of conservat ion, · wi th the addition of sui table expres. . sions for the slipfactor, two-phase friction multiplier and the heatdistribution parameter in the region of subcocled boiling. No special attention has been paid to the boiling boundary, the external system and estimated second order effects. For the solution procedure, a profitable application was made of the CSMP-program, developed by IBM, which facilitated the prograrnming of the integration process. The results of the computations were surprising owing to the close agreement with the experimental results with respect to the threshold-powers, the frequency, the destabilising effect of moderate subcooling and the influence of a non-uniform heatflux. Chapter 6 sunmarises the main conclusions to be drawn from the present study. The value of two-phase two-component measurements with the aim of transposition to boiling-water conditions is doubted. The equations of Neal and Zuber for the void propagation velocity are discussed. The influence of a non-uniform heatflux on the system stability·is reviewed and its consequence is stated for dynamic boiling water experiments and for the anticipated stability of steamboilers in general and of boiling water reactors in particular. Where it was possible and profitable chapters and parts of it have been completed with conclusive remarks. Note: A srnall part of the experiments to be reported here is similar to the experiments described by Spigt (S 2). Camparisen between both sets of results is impossible as void-fraction is concerned (see eh. 2.2.). Other results can 14 deviate some per cent. owing to the use of different heatilJ,g elements.
13 citations
"Non-linear dynamics of a two phase ..." refers background in this paper
...Djikman (1971) found that cosine flux stabilizes the flow....
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TL;DR: In this paper, a digital model based on conservation of mass, energy and momentum was developed to support the phase relationship between void fraction and inlet flow rate in a natural circulation boiling water channel, and the effect of a sine-shaped heat flux on the stability of the system.
Abstract: A digital model was developed based on conservation of mass, energy and momentum. The equations were supplemented by suitabe correlations for slip factor, two-phase friction multiplier and subcooled boiling parameter. The CSMP program of IBM was used for the solution. The purpose of the model was to support a number of experiments performed to study the phase relationship between void fraction and inlet flow rate in a natural circulation boiling water channel, and the effect of a sine-shaped heat flux on the stability of the system. The qualitative effects were represented very well and the quantitative agreement with experimental results was promising.
7 citations
01 Nov 1964
3 citations
"Non-linear dynamics of a two phase ..." refers background in this paper
...Stenning and Veziroglu (1963) were the first to physically explain the phenomena of DWO....
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