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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Dissertation•
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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...regions of the channel characteristic curve [158]....
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TL;DR: In this paper, the impact of axial power and void distribution on the stability of boiling water Reactor (BWR) systems was analyzed and a turning point bifurcation was detected.
Abstract: Bifurcation analyses of the impact of the void distribution parameter C{sub 0} and the axial power profile on the stability of boiling water reactors (BWRs) are reported. Bifurcation characteristics of heated channels (without nuclear feedback) appear to be very sensitive to the axial power profile. A turning point bifurcation was detected for a (symmetrically) peaked axial power profile. This kind of bifurcation does not occur for a uniformly heated channel.Both supercritical and subcritical Hopf bifurcations were encountered in a (nuclear-coupled) reactor system, depending on the strength of the void reactivity feedback. Subcritical bifurcations become less likely to occur as C{sub 0} is significantly larger than unity. In BWRs with a strong nuclear feedback, the oscillation amplitude of limit cycles caused by a supercritical bifurcation is very sensitive to both C{sub 0} and the axial power profile.
14 citations
TL;DR: In this article, a model to study two-phase flow instabilities taking into account general parameters from real systems is proposed and the stability influence of external parameters such as the fluid inertia and the presence of compressible gases in the system is analyzed.
Abstract: The most common kind of static and dynamic two-phase flow instabilities namely Ledinegg and density wave oscillations are studied. A new model to study two-phase flow instabilities taking into account general parameters from real systems is proposed. The stability influence of external parameters such as the fluid inertia and the presence of compressible gases in the system is analyzed. High-order oscillation modes are found to be related with the fluid inertia of external piping. The occurrence of high-order modes in experimental works is analyzed with focus on the results presented in this work. Moreover, both inertia and compressibility are proven to have a high impact on the stability limits of the systems. The performed study is done by modeling the boiling channel using a one dimensional equilibrium model. An incompressible transient model describes the evolution of the flow and pressure in the non-heated regions and an ideal gas model is used to simulate the compressible volumes in the system. The use of wavelet decomposition analysis is proven to be an efficient tool in stability analysis of several frequencies oscillations.
12 citations
TL;DR: In this paper, the effects of different non-uniform axial heat flux profiles (found to be a key factor) on the aforesaid nonlinear stability behavior of the system are shown which is missing in existing studies.
Abstract: A few nonlinear dynamics (Hopf bifurcations and limit cycles) of density wave oscillations of a two-phase flow channel are studied in this work. The effects of different non-uniform axial heat flux profiles (found to be a key factor) on the aforesaid non-linear stability behavior of the system are shown which is missing in existing studies. The novel outcomes of this study are presented by the change in Hopf bifurcation phenomena as the heat flux pattern and other design parameters are varied even though the total heat rate remains same. The innovative aspect of this study can be explained by the use of a nodalized reduced order model (NROM). The time-dependent spatial variations of the flow variables (enthalpy and quality) are approximated in-line to the provided heat flux profiles and corresponding steady state solutions. On one hand, this scheme is capable of analyzing both linear and nonlinear stability characteristics of the system over a wide range of axial heat flux profiles. On the other hand, in this model the node positions are defined in such a way that they vary automatically during the system evolution as per the provided heat flux profile. Hence, it acts as an auto-adaptive scheme and helps to minimize the computational effort required to achieve significant accuracy to capture the state variable variations. A few unique findings of the present work are as follows: i) although non-uniform axial heat flux is more linearly stable compared to uniform axial heat flux profile, it is less non-linearly stable. ii) single humped axial heat profiles are more linearly stable than double humped profiles, but they become less non-linearly stable.
12 citations
TL;DR: In this article, the authors report uncontrolled behavior and controlled behavior of boiling flow in a mini-channel excited by step perturbation and periodic perturbations and demonstrate the performance of nonlinear model predictive control (NMPC) in reducing the chaotic oscillations.
Abstract: The present study reports uncontrolled behavior and controlled behavior of boiling flow in a mini-channel excited by step perturbation and periodic perturbation. It also demonstrates the performance of nonlinear model predictive control (NMPC) in reducing the chaotic oscillations. The control performance significantly varies when the control region is changed from the supercritical bifurcation (stable periodic orbit) region at high inlet subcooling to the subcritical bifurcation (unstable periodic orbit) region at low inlet subcooling. The study also includes the effect of forcing amplitude and forcing frequency on the uncontrolled and controlled dynamics of the channel.
11 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.
5,239 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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