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: In this article, the authors reported bubble dynamics in a subcooled flow boiling in a vertical boiling channel based on numerical simulation using four-equation drift flux model (DFM4).
Abstract: In this article, we report bubble dynamics in a subcooled flow boiling in a vertical boiling channel based on numerical simulation using four-equation drift flux model (DFM4). The focus of this work is to validate the bubble phenomena computed by DFM4 in high inlet subcooling, low heat fluxes and low flow rates. The computed bubble parameters such as bubble growth time tg, bubble waiting time tw, bubble departure diameter Dd based on the available correlations were compared with the experimental results obtained from our boiling loop. We also propose an analytical solution for tg and Dd from the asymptotic growth kinetics and the force balance at the time of bubble departure. We have explained some important insights of the axial variation of bubble phenomena under varying heat flux. Unal correlation for bubble departure works extremely well to a highly subcooled boiling in low heat fluxes and low flows.
10 citations
TL;DR: In this paper, the effect of non-uniform axial heating on density wave oscillations of a two-phase flow channel is studied and the model behaves as an auto-adaptive scheme, where the mesh sizes between consecutive nodes automatically vary during the system evolution according to the provided shape of heat flux.
Abstract: The effect of non-uniform axial heating on density wave oscillations of a two-phase flow channel is studied. This study becomes relevant in view of designing and modelling of nuclear reactor heat removal systems as several reactor fuel rods (Boiling Water Reactors) exhibit non-uniform axial heat flux profiles. In this work a Nodalized Reduced Order Model (NROM) has been developed The unique feature of the model is that, using this model the stability behaviour of two-phase flow systems with a wide range of axial heat flux variations (single humped and double humped) can be studied. In addition to these, the model behaves as an auto-adaptive scheme, where the mesh sizes between consecutive nodes automatically vary during the system evolution according to the provided shape of axial heat flux. This helps to minimize the computational effort required to achieve significant accuracy to capture the state variable variations. The effects of single humped and various double humped axial heating profiles on local bifurcation have been studied. It has been observed in this study that in addition to the linear stability characteristic the nonlinear stability behaviour (depicted by the position of generalized Hopf points and occurrences of limit cycles) also changes significantly as the axial heat flux changes from uniform to non-uniform even though the total heat rate provided remains same. Moreover, a generalized description of the cause due to which these changes occur in stability behaviour of the system is also presented.
9 citations
TL;DR: In this article, the singularity theory and D-partition method are combined to determine the conditions under which a surge tank is introduced up stream of a boiling channel and the system shows sustained oscillations having a low frequency.
Abstract: Sustained oscillations have been experimentally observed when a surge tank is introduced up stream of a boiling channel. This dynamic state arises from an instability of the steady state of the system. These low-frequency oscillations are called pressure drop oscillations to distinguish them from the high-frequency density wave oscillations of boiling channels under constant pressure drop. The conditions under which these oscillations arise are that the steady-state operating point should be in the negative slope region of the boiling channel pressure drop characteristic and in the positive-slope region of the system pressure drop characteristic. The steady state should also be unique in the system pressure drop characteristic i.e. there should be no coexisting steady states. Under these conditions when this state is dynamically unstable the system shows sustained oscillations having a low-frequency. In this paper we discuss how the parameter values where these conditions are satisfied can be determined analytically and elegantly using results from the D-partition method and the singularity theory. The former allows us to determine regions in parameter space where an operating point is stable while the latter allows us to obtain different regions in parameter space where the bifurcation diagrams are qualitatively different. The superposition of results of these theories is used to determine parameter values where pressure drop oscillations occur. The predictions are verified with numerical simulations of the original nonlinear model for a horizontal channel.
9 citations
TL;DR: In this paper, the authors compared the heat transfer film coefficients predicted by the chaotic prediction method with those obtained from the experimental measurements and found that the chaotic forecast is effective for the estimations of the heat-transfer characteristics of the two-phase flow boiling system and may be a potential tool for the effective thermo-fluid control for such evaporators.
Abstract: The time series of heat-transfer film coefficient for an evaporator with a vapor–liquid two-phase boiling flow were forecasted by the chaotic prediction or the local nonlinear short-term prediction. The forecasting method was based on the phase space reconstruction theory and a mathematic model in the form of datum driving had been developed to carry out the prediction. The signals of heat-transfer film coefficients predicted by this method were compared with those obtained from the experimental measurements. Different from the previous work, the comparisons in this work were done both from the point of view of the time trajectory and from the point of view of macroscopic or general characters. Besides the time-averaged statistics characteristics, such as the average value, average deviation and standard deviation, the power spectrum, phase plane map and chaotic invariants including the correlation dimension and Kolmogorov entropy had been calculated and been compared for the time series obtained both from the experimental measurement and from the forecast. The comparison for each parameter between the value calculated from the time series measured and that estimated from the time series forecasted by the chaotic forecast method shows satisfactory agreement. The limited length in time with an accuracy prediction indicates that the system is chaotic. The agreement of the comparison of the general parameter indicates that the chaotic prediction is effective for the estimations of the heat transfer characteristics of the two-phase flow boiling system and that the chaotic prediction method may be a potential tool for the effective thermo-fluid control for such evaporators.
9 citations
TL;DR: In this paper, Density wave oscillations (DWO) and Ledinegg instabilities are analyzed for boiling and condensing systems in a single tube, based on a numerical model solved with a least squares spectral element method which is characterized by negligible numerical diffusion and high accuracy.
Abstract: In this work Density Wave Oscillations (DWO) and Ledinegg instabilities are analyzed for boiling and condensing systems in a single tube. The analysis is based on a numerical model solved with a least squares spectral element method which is characterized by negligible numerical diffusion and high accuracy. Stability limits are constructed and compared with available correlations. The analysis is extended to sub-cooled, saturated and over-heated inlet conditions. Finally a discussion regarding the occurrence of these phenomena in condensing systems is presented.
8 citations
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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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