Abstract
The essentials of the theory of fractals and chaos are reviewed and then applied to the problem of density-wave oscillations in a boiling channel. It is shown that these mathematical techniques give valuable new insight into nonlinear thermal-hydraulic oscillations.
Zusammenfassung
Die entscheidenden Formeln der Fraktal- und Chaostheorie wurden nochmals aufgezeigt und anschließend auf das Problem der zeitlich veränderten Dichtewellen in einem Siedekanal angewandt. Es wurde gezeigt, daß beide mathematische Verfahren einen wertvollen neuen Einblick in nicht lineare thermisch-hydraulische Schwingungen geben.
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Abbreviations
- A x−s :
-
Heated channel cross sectional area
- A R :
-
Riser cross sectional area
- c pf :
-
Liquid specific heat
- D :
-
Channel diameter
- D c :
-
Correlation dimension
- E :
-
Energy
- f :
-
Friction coefficient
- g :
-
Gravity
- h :
-
Specific enthalpy
- h fg :
-
Latent heat of vaporization
- h f :
-
Liquid enthaly
- K :
-
Loss coefficient
- L :
-
Channel length
- M :
-
Mass
- M 2 :
-
Two-phase mass in the heated channel
- N s :
-
Number of nodes in the subcooled region
- N R :
-
Number of nodes in the riser
- P H :
-
Heated perimeter
- p :
-
Pressure
- Δp :
-
Pressure drop
- q″:
-
Heat flux
- q :
-
Total power
- t :
-
Time
- u :
-
Velocity
- v f :
-
Specific voluje of the liquid
- v fg :
-
Liquid to vapor specific volume difference
- w :
-
Mass flow rate
- z :
-
Space variable
- β :
-
Liquid thermal expansion coefficient, − 1/ϱ ∄ϱ/∄T
- δ x :
-
Perturbation,x(t)−x 0
- ϱ :
-
Density
- ϱ f :
-
Liquid density
- Ω :
-
Characteristic frequency
- λ :
-
Boiling boundary
- a :
-
Acceleration head
- ch:
-
Channel
- D :
-
Downcomer
- e :
-
Channel exit
- ext:
-
External
- f :
-
Friction head
- i :
-
Channel inlet
- I :
-
Inertial head
- g :
-
Gravity head
- n :
-
nth subcooled node
- ref:
-
Reference value
- R :
-
Riser
- 2φ :
-
Two-phase
- 0:
-
Steady-state
- \(N_{p{\text{ch}}} = \frac{{q_0 }}{{w_0 }}\frac{{v_{fg} }}{{h_{fg} v_f }}\) :
-
(Phase change number)
- \(N_{{\text{sub}}} = \frac{{(h_f - h_i )}}{{h_{fg} }}\frac{{v_{fg} }}{{v_f }}\) :
-
(Subcooling number)
- \(Fr = \frac{{u_{i_c }^2 }}{{gL_H }}\) :
-
(Froude number)
- Λ =fL/2D :
-
(Friction number)
- \(b = \frac{{\beta h_{fg} v_f }}{{c_{pf} v_{fg} }}\) :
-
(Thermal expansion number)
References
Feder, J.: Fractals. New York: Plenum Press 1988
Mandelbrot, B. B.: Fractals, form, chance and dimension. W. H. Freeman, San Francisco, Ca., 1977
Barnsley, M.: Fractals everywhere. New York: Academic Press Inc. 1988
Barnsley, M. F.; Devaney, R. L.; Mandelbrot, B. B.; Peitgen, H.-O.; Saupe, D.; Voss, R. F.: The science of fractal images. Berlin Heidelberg New York: Springer 1988
Moon, F. C.: Chaotic vibrations. New York: Wiley & Sons 1987
Packard, N. H.; Crutchfield, J. P.; Farmer, J. D.; Shaw, R. S.: Geometry from a time series. Philadelphia Review Letters 45 1980
Ben-Mizrachi, A.; Procaccia, I.: Characterization of experimental (noisy) strange attractors. Physical Review A. 29, No. 2, 1984
Dorning, J.; Nonlinear dynamics and chaos in heat transfer and fluid flow. AIChE Symposum Series-269, 85, 1989
Bergé, P.; Pomeau, Y.; Vidal, C.: Order within chaos. New York: Wiley & Sons 1984
Marsden, J.; McCracken, M.: The Hopf bifurcation and its applications. App. Math. Series Berlin Heidelberg New York: Springer 1976
Achard, J. L.; Drew, D. A.; Lahey, R. T., Jr.: The analysis of nonlinear density-wave oscillations in boiling channels. J. Fluid Mechanics 155 1985
Rizwan-Uddin; Dorning, J. J.: Some nonlinear dynamics of a heated channel. Nuc. Eng. & Des., 93 1986
Feigenbaum, M. J.: Qualitative universality for a class of nonlinear transformations, J. Statistical Physics, 19(1), 1978
Hassard, B. D.; Kazarinoff, N. D.; Wau, Y.-H.: Theory and applications of Hopf bifurcation, London Mathematical Society Lecture Note Series-41, Cambridge: University Press 1981
Thompson, J. M. T.; Stewart, H. B.: Nonlinear dynamics and chaos. New York: Wiley & Sons 1986
Lorenz, E. N.: Deterministic non-periodic flow. J. Atmos. Sci. 20. (1963)
Lahey, R. T.; Jr. & Moody, F. J.: The thermal-hydraulics of a boiling water nuclear reactor. ANS Monograph, 1977
Rizwan-Uddin; Dorning, J. J.: A chaotic attractor in a periodically forced two-phase flow system. Nucl. Sci. & Eng. 100 (1988) 393–404
LeCoq, G.: Steam generator dynamic instabilities simulation — experimental analysis and theoretical interpretation. Proceedings of the 3rd. Int. Topical Meeting on Nuclear Power Plant Thermal-Hydraulics and Operations. Seoul, Korea, 1988
Clausse, A.; Lahey, R. T., Jr.: An investigation of periodic and strange attractors in boiling flows using chaos theory. Proceedings of the 9th International Heat Transfer Conference, Jerusalem, Israel, 1990
Clausse, A.: Efectos no Lineales en Ondas de Densidad en Flujos bifasicos. PhD thesis. Instituto Balseiro, 8400 Bariloche, Argentina, 1986
Delmastro, D. G.: Influencia de la gravedad sobra la estabilidad de canales en ebullición. M. S. thesis, Instituto Balseiro, 8400 Bariloche, Argentina, 1988
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Dedicated Prof. Dr.-Ing. F. Mayinger's 60th birthday
t + =t/v
\(v{\text{ }} = {\text{ }}\frac{{N_{sub} }}{{N_{pch} }}\frac{L}{{u_{i_0 } }} = \lambda _0 /u_{i_0 }\)
M + =M/(ϱ f LA x−s)
z + =z/L
A + R =A R/A x−s
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Lahey, R.T. An application of fractal and chaos theory in the field of two-phase flow & heat transfer. Wärme- und Stoffübertragung 26, 351–363 (1991). https://doi.org/10.1007/BF01591668
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DOI: https://doi.org/10.1007/BF01591668