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Journal Article•DOI•

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

S Narayanan1, B. Srinivas1, S. Pushpavanam1, S. Murty Bhallamudi1•
Indian Institute of Technology Kanpur1
04 Dec 1997-Nuclear Engineering and Design (Elsevier BV)-Vol. 178, Iss: 3, pp 279-294
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.
Citations
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Journal Article•DOI•
Periodically-forced density wave oscillations in boiling flow at low forcing frequencies: Nonlinear dynamics and control issues

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Swapan Paruya1, Nababithi Goswami1, S. Pushpavanam2, Dipin S. Pillai2, Oinam Bidyarani1 •
National Institute of Technology, Durgapur1, Indian Institute of Technology Madras2
02 Feb 2016-Chemical Engineering Science
TL;DR: In this article, the authors investigate the effect of a periodic perturbation of total pressure drop on density wave oscillations (DWOs) in a boiling channel (BC) and propose a nonlinear model predictive controller (NMPC) to control the periodically-forced DWOs.
Abstract: In this paper, we investigate the effect of a periodic perturbation of total pressure drop on density wave oscillations (DWOs) in a boiling channel (BC). The focus is on the nonlinear dynamics and control of the DWO at low forcing frequencies ff+ for a range of inlet subcooling number Nsub. At a relatively high Nsub, the system exhibits frequency entrainment. Relying on the harmonic balance applied to the linearized model for DWOs, semi-analytical solutions can successfully explain the numerically-detected harmonic entrainment. The effect of ff+ and Nsub on the periodically-perturbed DWOs has been studied. We also propose a nonlinear model predictive controller (NMPC) to control the periodically-forced DWOs. The performance of NMPC was found to be improving on increasing Nsub and on decreasing ff+ in 0≤ff+<2fn+ (fn+ is natural frequency), except the regions of entrainment. Both the numerical simulations and the semi-analytical solutions consistently predict the control effort that also varies with ff+ and Nsub.

7 citations

Journal Article•DOI•
Stability monitoring tests using a nuclear-coupled boiling channel model

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Miguel Ceceñas-Falcón1, Robert M. Edwards1•
Pennsylvania State University1
01 Jul 2000-Nuclear Technology
TL;DR: In this article, the thermal hydraulics for boiling channels are modeled and coupled with neutron kinetics to analyze the nonlinear dynamics of the closed-loop system, and point kinetics is used to study corewide oscillations, and it couples two time-domain calculations, for the fundamental and first harmonic modes, to study out-of-phase oscillations.
Abstract: A new test platform for stability studies is presented that can be used to generate a power time series, which in turn may be used to validate the capability of boiling water reactor stability-monitoring algorithms. The thermal hydraulics for boiling channels are modeled and coupled with neutron kinetics to analyze the nonlinear dynamics of the closed-loop system, The model uses point kinetics to study core-wide oscillations, and it couples two time-domain calculations, for the fundamental and first harmonic modes, to study out-of-phase oscillations. The channel coolant flow dynamics is dominant in the power fluctuations observed by in-core nuclear instrumentation, and additive white noise is added to the solution for the channel flow in the thermal-hydraulic model to generate a noisy power time series. Autoregressive analysis performed with the computer-generated series agrees with the stability properties of the boiling channel. The operating conditions of the channel can be modified to accommodate a wide range of stability conditions.

6 citations

Journal Article•DOI•
Nonlinear Analysis of Chaotic Time Series in a Natural Circulation Boiling Loop

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Arnab Karmakar1, Swapan Paruya1•
National Institute of Technology, Durgapur1
05 Dec 2012-Industrial & Engineering Chemistry Research
TL;DR: In this article, chaotic flow oscillations observed in a natural circulation boiling loop are reported, where the periodic oscillations of wall temperature (thermal oscillations) initiate at a certain condition of heat flux and inlet subcooling in addition to geysering instability and pressure-drop oscillations.
Abstract: The present study reports chaotic flow oscillations observed in a natural circulation boiling loop. The periodic oscillations of wall temperature (thermal oscillations) initiate at a certain condition of heat flux and inlet subcooling in addition to geysering instability and pressure-drop oscillations. We observed that boiling regime changes from nucleate boiling to transition boiling as a result of decrease in inlet subcooling and increase in heat flux. The thermal oscillations are strongly coupled with pressure-drop oscillations. Nonlinear analysis of the time series of loop flow rate at various heater power and inlet subcooling have been carried out using statistical analysis, fast Fourier transform (FFT), time delay embedding for attractor reconstruction, autocorrelation, and correlation dimension. The analysis confirms that the oscillations are more chaotic at relatively low heater power and high inlet subcooling. The complexity of the oscillations strongly depends on boiling heat transfer regime. Our observations and analysis have been supported by other relevant experiments.

5 citations

Journal Article•DOI•
Research of two‐phase density wave instability in reactor core channels with rolling motion

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Wenxi Tian1, Qiang Lian1, Suizheng Qiu1, Guanghui Su1•
Xi'an Jiaotong University1
01 Jul 2020-International Journal of Energy Research

5 citations

Cites background from "Non-linear dynamics of a two phase ..."

  • ...As explained in Equation (4), ΔPf, ΔPg, ΔPa, ΔPl, in Figure 12, are frictional, gravitational, acceleration, and form pressure drop, respectively....

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  • ...The additional acceleration a is a vector composition of translational acceleration and rotational acceleration, which is shown as Equation (21). a=at +ω× ω× rð Þ+ β× r+2ω×u ð21Þ where the vector terms can be decomposed into the components along three axes (x, y, z), as shown in Equation (22). a= atx aty atz 2 64 3 75+ ωx ωy ωz 2 64 3 75× ωx ωy ωz 2 64 3 75× rx ry rz 2 64 3 75 0 B@ 1 CA+ βx βy βz 2 64 3 75× rx ry rz 2 64 3 75 + 2 ωx ωy ωz 2 64 3 75× ux uy uz 2 64 3 75 ð22Þ According to Equation (21), the additional force on the fluid Fa can be expressed in Fa =F t +Fce +F ta +Fco = −ρ at +ω× ω× rð Þð + β× r+ 2ω×uÞ ð23Þ where Ft is the translational inertial term, Fce is the centrifugal term, Fta is the tangential term, and Fco is the Coriolis term....

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  • ...For the onedimensional analysis along the channel, Equation (27) can be reduced to the form used in Ma et al.23 With the additional acceleration on fluid, dPadd can be derived by Equation (28) in which k is the unit vector of flow direction....

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  • ...The expression to formulate the void fraction in Chexal et al18 is given as Equation (5). α= β C0 + Vgj=j ð5Þ Combining the vapor volumetric fraction in Equation (6), the void fraction in Equation (5) could be transformed into Equation (7). β= 1 1+ 1−xð Þ=xð Þ+ ρg=ρl ð6Þ α= 1 C0 1+ 1−xð Þ=x ρg=ρl + Vgjρg=xG ð7Þ The expression of drift velocity for calculation of void fraction is given as: Vgj =1:41 ρl−ρg σg ρ2l 2 4 3 5 1=4 C1C2C3C4 ð8Þ The parameters Ci are given in detail in Chexal et al. 18 Its validity is verified for .01 ≤ α ≤ .95, 0.01 ≤ G ≤ 2100 kg/m2/s1, and 0.1 ≤ P ≤ 14.5 MPa under nonadiabatic conditions....

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  • ...Nsub =Npch−Xe vfg vfs ð31Þ For a given subcooling condition, after the threshold heating power triggering instability in the system is obtained in experiment or calculation, the corresponding heating power Q, flow rate w and other physical properties are substituted into Equations (29) and (30) to calculate Npch and Nsub....

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Journal Article•DOI•
Nonlinear forecast of heat transfer coefficient signal in fluidized bed evaporator

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Mingyan Liu1, Juan-ping Xue1, Ai-hong Qiang1•
Tianjin University1
01 Jan 2008-Chemical Engineering Research & Design
TL;DR: In this paper, a model of datum driving developed on the basis of reconstruction of phase space is used to forecast the evolution behavior of heat transfer coefficient in an evaporator with vapor-liquid-solid external natural circulation flow.
Abstract: The focus of this work is on the local nonlinear forecast of evolution behavior of heat transfer coefficient in an evaporator with vapor–liquid–solid external natural circulation flow. Time series of heat transfer coefficients are calculated from the measured signals of wall and fluid temperatures in the fluidized bed evaporator. The times series predicted by a model of datum driving developed on the basis of reconstruction of phase space, are compared to the obtained signals experimentally according to the time-averaged statistics characteristics, power spectral density functions, phase maps and invariants of chaotic attractor of evolution system including correlation dimension and Kolmogorov entropy. The predictions of average heat transfer coefficients show reasonable agreements, and those of the heat transfer coefficient fluctuations display tendency agreements. The time series predictions for vapor–liquid–solid flow boiling system is not as good as those for vapor–liquid flow boiling and gas–liquid bubble column systems from the point of view of the time trajectory, which indicates certain essential differences between these systems in fluctuation mechanism and nonlinear complexity.

4 citations

References
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Book•
Heat Transfer

[...]

J. P. Holman
01 Jan 1976

9,629 citations

Journal Article•DOI•
Measuring the Strangeness of Strange Attractors

[...]

Peter Grassberger1, Peter Grassberger2, Itamar Procaccia1•
Weizmann Institute of Science1, University of Wuppertal2
01 Oct 1983-Physica D: Nonlinear Phenomena
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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Book•DOI•
Finite Difference Methods in Heat Transfer

[...]

M. Necati Özişik1•
North Carolina State University1
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

Journal Article•DOI•
Proper choice of the time delay for the analysis of chaotic time series

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Wolfgang Liebert, H. G. Schuster1•
University of Kiel1
04 Dec 1989-Physics Letters A
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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Journal Article•DOI•
An Experimental Investigation of the Thermally Induced Flow Oscillations in Two-Phase Systems

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P. Saha1, Mamoru Ishii2, N. Zuber3•
General Electric1, Argonne National Laboratory2, Nuclear Regulatory Commission3
01 Nov 1976-Journal of Heat Transfer-transactions of The Asme
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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Review of two-phase flow instability

[...]

01 Jul 1973-Nuclear Engineering and Design
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The analysis of nonlinear density-wave oscillations in boiling channels

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01 Jun 1985-Journal of Fluid Mechanics
Jean-Luc Achard, Donald A. Drew, Richard T. Lahey
A Chaotic Attractor in a Periodically Forced Two-Phase Flow System

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01 Dec 1988-Nuclear Science and Engineering
Rizwan-uddin, J. J. Doming
A Review of two-phase flow dynamic instabilities in tube boiling systems

[...]

01 Feb 2008-International Journal of Heat and Mass Transfer
Sadik Kakac, B. Bon
Fundamental and Higher-Mode Density-Wave Oscillations in Two-Phase Flow

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01 May 1972-Journal of Heat Transfer-transactions of The Asme
G. Yadigaroglu, A. E. Bergles