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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01 Jan 2017
TL;DR: In boiler construction, depending on the fuel used, we distinguish between conventional systems (those that use fossil fuels like oil, gas, or coal) and nuclear plants in which nuclear fuels such as235U are used as discussed by the authors.
Abstract: In boiler construction—depending on the fuel used—we distinguish between “conventional” systems (those that use fossil fuels like oil, gas, or coal) and nuclear plants in which nuclear fuels such as235U are used. The following summary of boiler systems, however, only addresses fossil fuel-fired plants. A brief overview on the main features of reactor theory and nuclear fission, as well as on the construction of nuclear reactors is given in Thomas (1975), Ziegler (1983), Ziegler (1984), Ziegler (1985), Straus (1992), Weston (2007), Kok (2009), Todreas and Kazimi (2012), or Oka (2014).
3 citations
TL;DR: In this paper, the authors compute the dynamics of a spherical vapor-bubble in an infinite pool of subcooled water during bubble collapse using a semi-analytical method.
Abstract:
In this work, we compute the dynamics of a spherical vapor-bubble in an infinite pool of subcooled water during bubble collapse using our semi-analytical method. The main contribution of this work is to bring out the dynamics of nonmonotonic bubble collapse describing heat transfer characteristics and nonlinear dynamics. The dynamics shows the variation of radius with time for collapsing vapor bubble at different subcooling ΔTsub of 1.40 K to 35 K. The present approach accurately determines the bubble radius decreasing with time and has been compared with our experimental results, the experiment from literature, the other theories, and correlations. As it is noted that the literature lacks steady-state analysis of oscillating bubble collapse, we also report the steady-state analysis and the bifurcation analysis of bubble collapse at a pressure of 1.0 atm to check the stability of bubble collapse. The effect of ΔTsub and initial bubble radius R0 on dynamics of bubble collapse has been analyzed. The collapse of big bubbles involves with the bubble oscillations because of a large contribution of liquid inertia and the collapse of very small bubbles essentially occurs in heat transfer regime.
2 citations
TL;DR: In this article, a systematic method by using the NARMA model, which takes advantage of both the ANN and ARMA, is developed for the identification of the dynamics of the Vermont Yankee BWR with the reactor noise.
Abstract: For the identification of the dynamics of the Vermont Yankee BWR with the reactor noise, different parametric models have been tested. The widely used ARMA model is unable to identify the nonlinearity in the noise data. A systematic method by using the NARMA model, which takes advantage of both the ANN and ARMA, is developed. Comparisons are made between the identification results with ARMA and NARMA model. The advantages of identification with NARMA model over ARMA model are demonstrated. The linear-kernels of the identified NARMA models are extracted so that the natural frequency, damping ratio and time constants of the BWR are obtained. The values of those characteristics are well corresponded with the eigenvalues calculated by the differential equations of the Vermont Yankee BWR. The damping ratio with negative value is found to be a criterion for the existence of limit-cycle, which can be seen from the impulse response on the (X t , X t-1 ) plane, in stable nonlinear system.
1 citations
01 Jan 2021
TL;DR: In this paper, the authors show that the conventional approach of identifying the instability thresholds does not hold good to determine the global stability behavior of the system and propose the need for redefining the criteria and the approach to identify the instability threshold experimentally.
Abstract: Two-phase flow instabilities and in particular density wave oscillations (DWOs) are unwanted in boiling, condensation, and other flow boiling systems as it can cause severe deterioration in the performance or even damage the system. For decades, efforts have been focused on designing equipment or processes that operate far from the conditions where the two-phase flow instabilities are present. And hence multiple studies in particular experiments were carried out to identify the characteristics of the instability thresholds during the DWOs. In this work, we show that the conventional approach of identifying the instability thresholds does not hold good to determine the global stability behavior of the system. This includes identifications of the limit cycle oscillations and the Hopf bifurcation across the instability thresholds. And hence, this study postulates the need for redefining the criteria and the approach to identify the instability thresholds experimentally.
TL;DR: In this article, the authors developed a numerical model which simulates the hydrodynamics of forced and natural circulation boiling water nuclear reactor loops and demonstrates its capability to analyse density wave instability of the reactor undergoing in-phase and out-of-phase modes of oscillations.
Abstract: The present work focuses on the development of a numerical model which simulates the hydrodynamics of forced and natural circulation boiling water nuclear reactor loops and demonstrates its capability to analyse density wave instability of the reactor undergoing in-phase and out-of-phase modes of oscillations. The numerical algorithm integrates a compressible and distributed fluid flow model used for simulating the two-phase flow phenomenon in the channels of the reactor core with an incompressible lumped parameter model which accounts for sub-cooled water flow in various ex-core and recirculation components of the reactor loop. The algorithm used in the present hybrid model explicitly addresses the uncertainties related to boundary conditions existing in the literature to simulate parallel channel instability analysis for forced and natural circulation boiling reactor loops. Extensive numerical experimentations are performed to search the stability thresholds for both the forced and natural circulation reactor loops undergoing in-phase and out-ofphase instability modes.
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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.
5,239 citations
"Non-linear dynamics of a two phase ..." refers methods in this paper
...The attractor is reconstructed using the time delay embedding method described in Grassberger and Procacia (1983), Leibert and Schuster (1991)....
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...The attractor is reconstructed using the time delay embedding method described in Grassberger and Procacia (1983), Leibert and Schuster (1991). We now discuss how the method can be used to characterise the dynamic behaviour of the system using velocity time series data....
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...The attractor is reconstructed using the time delay embedding method described in Grassberger and Procacia (1983), Leibert and Schuster (1991). We now discuss how the method can be used to characterise the dynamic behaviour of the system using velocity time series data. It also enables us to determine the minimum number of variables required to describe the system completely. The reconstructed attractor is obtained from the following steps 1. The time series 6in(t) is generated by simulations. The time interval of integration is 0.005 (in dimensionless time units). This time series is viewed as experimental data obtained by sampling the inlet velocity at the above time period. The system can be characterised quantitatively if we can determine the embedding dimension d and delay time Td. Optimum values of these variables is essential for the successful computation of the attractor dimension g. 2. The time delay Td is estimated in terms of the sampling time using the method of Leibert and Schuster (1991). A sample of 5000 points (after the initial transients have died down) was used to determine the delay time....
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20 Jul 2017
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
636 citations
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.
310 citations
"Non-linear dynamics of a two phase ..." refers methods in this paper
...The attractor is reconstructed using the time delay embedding method described in Grassberger and Procacia (1983), Leibert and Schuster (1991)....
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...The time delay Td is estimated in terms of the sampling time using the method of Leibert and Schuster (1991)....
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...This is called the upwind difference scheme (Ozicik (1994))....
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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.
149 citations
"Non-linear dynamics of a two phase ..." refers background in this paper
...Saha et al. (1976) studied experimentally the effect of various parameters like inlet sub-cooling, inlet restriction, exit restriction on the stability of the system to DWO....
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