Wärme - und Stoffübertragung

, Volume 13, Issue 4, pp 275–286 | Cite as

Heat flux-flow coupling effects in the stability of vapour generators

  • C. D. Rakopoulos
  • A. A. El-Shirbini
  • W. Murgatroyd
Article

Abstract

A study has been carried out to investigate the dynamics of two-phase flow in vapour generators, with special reference to the effect of flux-flow couplings on density wave instability.

A distributed parameter time-varying theoretical analysis of the conservation equations, together with the constitutive laws of two-phase flow, has been derived. The analysis takes into account the dynamics of the heater wall, revealing the predominant role of the heat flux-flow couplings on the stability of the system. A digital computer program has been written and developed to implement the theoretical analysis. Results from the analysis are used for comparison with the available data and the effect of the flux-flow couplings on instability is demonstrated and discussed. These results are also used to explain the dynamic behaviour of a specially constructed “Arcton-113”1 closed loop comprising a helical, electrically heated, once-through vapour generator in a parallel channel arrangement with high thermal capacitance. Kopplungseffekte zwischen Wärmefluß und Strömung bei der Stabilität von Dampferzeugern

Keywords

Coupling Effect Digital Computer Density Wave Predominant Role Parallel Channel 

List of Symbols

A

cross-sectional area (m2)

a

thermal diffusivity (m2/s)=k/ρ c

b

exponent in heat transfer correlation

c

specific heat (J/kg°C)

d

tube diameter (m)

f

frequency (Hz)

G

mass velocity (kg/m2s)

h

specific enthalpy (J/kg)

hcn

convective heat transfer coefficient (W/m2°C)

hfg

heat of evaporation (J/kg)

j

imaginary unit=√−1

k

thermal conductivity (W/m°C)

K

experimental orifice coefficient

P

pressure (N/m2)

Q

heating power (W)

Qv

heating power per unit volume (W/m3)

q

parameter defined as: (vg−vf) Qv/hfg

Q′

heating power per unit length (W/m)

Q″

heat flux (W/m2)

r

tube radius (m)

r*

parameter defined as:\(e^{\gamma q\tau _2 }\)

R

coil radius (m)

s

Laplace variable: σ + jω

T

temperature (°C)

t

time (s)

U

velocity (m/s)

u

velocity (m/s)

v

specific volume (m3/kg)

W

mass flowrate (kg/s)

y

distance (m)

z

axial coordinate (m)

Greek Symbols

γ

slip coefficient

Δ

difference

δ

small perturbation

ξ

heated perimeter (m)

ρ

density (kg/m)

τ

time constant (s)

ω

circular or angular frequency (rad/s)

Δh

inlet subcooling (J/kg)

\(\phi ^2\)

measure of signal-to-noise ratio

σx

standard deviation (v)

ψ2

power content (statistical (v2))

Subscripts

b

boiling boundary

s

superheat boundary

1

subcooled single-phase region

3

superheat single-phase region

2

two-phase region

f

liquid phase

g

vapour phase

fg

liquid-vapour phase

w, m

wall (metal)

nr

normalised

MP

maximum peak

Superscripts

o

steady-state

Laplace transform of perturbed quantity

lDimensionless Groups

Nu

Nusselt number (h.d/k)

Pr

Prandtl number (c.μ/k)

Re

Reynolds number (p. U.d/μ)

Kopplungseffekte zwischen Wärmefluß und Strömung bei der Stabilität von Dampferzeugern

Zusammenfassung

Es wurden Untersuchungen über die Dynamik der Zweiphasenströmung in Dampferzeugern durchgeführt. Dabei interessierte vor allem der Einfluß einer Kopplung zwischen Wärmefluß und Strömung auf die Dichtewelleninstabilität. Eine theoretische Berechnungsmethode mit verschiedenen zeitabhängigen Parametern für die Erhaltungsgleichungen und die bekannten Gesetze für die Zweiphasenströmung wurde abgeleitet. Diese Methode berücksichtigt die Dynamik der Heizwand, die Aufschluß gibt über den entscheidenden Einfluß der Kopplung von Wärmefluß und Strömung auf die Stabilität des Systems.

Ein Computerprogramm wurde für die theoretische Berechnung geschrieben und entwickelt. Ergebnisse dieser Berechnung werden mit anderen, zur Verfügung stehenden Werten verglichen. Dabei wird der Einfluß der Kopplungseffekte auf die Instabilität gezeigt und erläutert. Schließlich dienen die Ergebnisse dazu, das dynamische Verhalten eines speziell konstruierten, geschlossenen “Arcton-113”2 Kreislaufes zu beschreiben, der aus einem schraubenförmigen, elektrisch beheizten Verdampferrohr in einem parallelen Kanal mit hoher Wärmekapazität besteht.

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References

  1. 1.
    Boure, J.A.; Bergles, A.E.; Tong, L.S.: Review of two-phase flow instability, Nuclear Eng. & Design, 25 (1973) 165Google Scholar
  2. 2.
    Potter, R.: A review of two-phase flow instability of boiler dynamics, Boiler Dynamics & Control in Nuclear Power Stations, B.N.E.S. (London), 13 (1973)Google Scholar
  3. 3.
    Neal, L.G.; Zivi, S.M.; Wright, R.W.: The mechanisms of hydrodynamic instabilities in boiling systems, Symp. on Two-Phase Flow Dynamics, EUR-4288e, Eindhoven, No. 8.1, (1967) 957Google Scholar
  4. 4.
    Moxon, D.: Stability of once-through steam generators, ASME Paper No. 73-WA/HT-24 (1973)Google Scholar
  5. 5.
    Verheugen, A.N.J.; Dijkman, F.J.M.; Lamein, H.J.; Tong, L.S.: The influence of the response characteristics of the heating rod on the stability and burn-out characteristics of a boiling channel, Symp. on Two-Phase Flow Dynamics, EUR-4288e, Eindhoven, 7.3 (1967) 841Google Scholar
  6. 6.
    Abi-Zadeh, D.: A non-linear digital simultation of the dynamic behaviour of a steam generator, Ph. D. Thesis, University of London, 1976Google Scholar
  7. 7.
    Yadigaroglu, G.; Bergles, A. E.: Fundamental and higher-mode density wave oscillations in two-phase flow, J. Heat Transfer, Trans. ASME (1972) 189Google Scholar
  8. 8.
    Rakopoulos, C.D.: The dynamics of two-phase flow in vapour generators, Ph.D. Thesis, University of London, 1976Google Scholar
  9. 9.
    Davies, A.L.; Potter, R.: Hydraulic stability: An analysis of the causes of unstable flow in parallel channels, Symp. on Two-Phase Flow Dynamics, EUR-4288e, Eindhoven, No. 9.3 (1967) 1225Google Scholar
  10. 10.
    Shinners, S.M.: Modern control system theory and application, Addison-Wesley Publishing Company Incorporated, 1972Google Scholar
  11. 11.
    DiStefano III, J.J.; Stubberud, A. R.; Williams, I. J.: Feedback and control systems, Schaum' s Outline Series, McGraw-Hill Book Company, 1967Google Scholar
  12. 12.
    Seban, R.A.; McLaughlin, E.F.: Heat transfer in tube coils with laminar and turbulent flow, Int. J. Heat & Mass Transfer, 6 (1963) 387Google Scholar
  13. 13.
    Eckert, E.R.G.; Drake, Jr, R.M.: Heat and mass transfer, McGraw-Hill Book Company Incorporated, 1959Google Scholar
  14. 14.
    McLachlan, N.W.: Bessel functions for engineers. Oxford University Press 1961Google Scholar
  15. 15.
    Abi-Zadeh, D.: Private communication, Imperial College, University of London, 1975Google Scholar
  16. 16.
    Sana, P.; Ishil, M.; Zuber, N.: An experimental investigation of the thermally induced flow oscillations in two-phase systems, ASME Paper No. 75-WA/HT-6, 1975Google Scholar
  17. 17.
    Rakopoulos, C.D.; El-Shirbini, A.A.; Murgatroyd, W.: An experimental investigation into the dynamics of vapour generators. Presented at the 6th Int. Heat Transfer Conf., Toronto, Canada, 1978Google Scholar

Copyright information

© Springer-Verlag 1980

Authors and Affiliations

  • C. D. Rakopoulos
    • 1
  • A. A. El-Shirbini
    • 1
  • W. Murgatroyd
    • 1
  1. 1.Department of Mechanical EngineeringImperial College of Science and TechnologyLondon

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