Impact of system pressure on the characteristics of stability boundary for a single-channel flow boiling system

  • Deepraj Paul
  • Suneet SinghEmail author
  • Surendra Mishra
Original Paper


The two-phase flow boiling channel as a nonlinear dynamical system has very interesting features. Parametric stability analysis of a flow boiling thermal hydraulic system has been extensively studied in the past. Linear stability analysis provides a stability boundary which is a demarcation between stable and unstable region. The studies on the impact of system pressure on the stability boundary are limited. Moreover, the variation of subcritical and supercritical Hopf region on stability boundary with respect to pressure is not available in the literature. Nonlinear dynamics is used here to characterize the stability boundary(Hopf curve) for a single-channel flow boiling thermal hydraulic system. In the current work, focus is on the effect of the system pressure on the location of the generalized Hopf (GH) point over the Hopf curve which divides subcritical and supercritical Hopf region. The disappearance of the GH point with increasing system pressure, such that the stability boundary is completely subcritical, in the operating parameter range is one of the interesting features observed. The complete subcritical boundary is identified by noting that the first Lyapunov coefficient is always positive on the stability boundary for higher system pressures. The subcritical Hopf bifurcation indicates the presence of unstable limit cycle in the (linearly) stable region which is of prime concern as finite perturbation in this region leads to system instability. Numerical simulations are carried out around the stability boundary to verify its characteristics.


Bifurcation Subcritical Hopf Supercritical Hopf Generalized Hopf Thermal hydraulics Mathematical modelling 

List of symbols

Greek symbols

\(\delta \)

Dirac delta function

\(\lambda \)

Boiling boundary (m)

\(\lambda ^+\)

Non-dimensional boiling boundary \((\frac{\lambda }{L_H})\)

\(\Lambda _{1\phi }\)

Single-phase friction number \(\left( \frac{f_{1\phi }L_H}{2D_H}\right) \)

\(\Lambda _{2\phi }\)

Two-phase friction number \(\left( \frac{f_{2\phi }L_H}{2D_H}\right) r\)

\(\rho \)

Density \(({\hbox {kg/m}}^3)\)

\(\rho ^+\)

Non-dimensional density \(\left( \frac{\rho }{\rho _f}\right) \)



Steady state


Channel exit






Heated channel


Channel inlet


nth node in single-phase region

Other symbols

\(\Delta P\)

Pressure drop (Pa)

\(\Delta P^+\)

Non-dimensional pressure drop \(\left( \frac{\Delta P}{\rho _f u_\mathrm{s}^2}\right) \)


Area \(({\hbox {m}}^2)\)


Liquid constant pressure specific heat \(({\hbox {J/kg \,K}})\)


Diameter (m)


Friction factor

\(f_{1\phi }\)

Single-phase friction factor

\(f_{2\phi }\)

Two-phase friction factor


Froude number \(\left( \frac{u_\mathrm{s}^2}{g L_H}\right) \)


Gravity acceleration \(({\hbox {m}}^2/{\hbox {s}})\)


Enthalpy \(({\hbox {kJ/kg}})\)


Non-dimensional enthalpy \(\left( \frac{h - h_f}{h_f}\right) \)


Resistance coefficient


Length (m)


Non-dimensional length \((L/L_H)\)


Mass (kg)


Non-dimensional mass \(\left( \frac{M}{\rho A_H L_H}\right) \)


No. of nodes in single-phase region


Thermal expansion number \(\left( \frac{\beta h_{fg} \rho _g}{C_{pf}\rho _{fg}}\right) \)


Phase change number \(\left( \frac{Q}{\rho _f A_H u_\mathrm{s}}\frac{\rho _{fg}}{h_{fg}\rho _g}\right) \)


Pressure (Pa)


Heating power (W)


Steady-state heat flux \(({\hbox {W/m}}^2)\)


Time (s)


Velocity (m/s)


Non-dimensional velocity \((u/u_\mathrm{s})\)


Steady-state velocity (m/s)


Axial coordinate (m)


Non-dimensional axial coordinate \(\left( \frac{z}{L_H}\right) \)


Equilibrium quality


Non-dimensional time \(\left( \frac{t}{t_\mathrm{ref}}\right) \)


Subcooling number \(\left( \frac{h_i - h_f}{h_{fg}\rho _g}\rho _{fg}\right) \)


Timescale \(\left( \frac{L_H}{u_\mathrm{s}}\right) \)


Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


  1. 1.
    Alobaid, F., Mertens, N., Starkloff, R., Lanz, T., Heinze, C., Epple, B.: Progress in dynamic simulation of thermal power plants. Prog. Energy Combust. Sci. 59, 79–162 (2017). CrossRefGoogle Scholar
  2. 2.
    Boure, J., Bergles, A., Tong, L.: Review of two-phase flow instability. Nucl. Eng. Des. 25(2), 165–192 (1973). CrossRefGoogle Scholar
  3. 3.
    Chatoorgoon, V.: Sports: a simple non-linear thermalhydraulic stability code. Nucl. Eng. Des. 93(1), 51–67 (1986). CrossRefGoogle Scholar
  4. 4.
    Clausse, A., Lahey, Jr., R.: An investigation of periodic and strange attractors in boiling flows using chaos theory. In: Heat Transfer 1990. Proceedings of the Ninth International Heat Transfer Conference (1990)Google Scholar
  5. 5.
    Dhooge, A., Govaerts, W., Kuznetsov, Y.A.: MATCONT: a MATLAB package for numerical bifurcation analysis of ODEs. ACM Trans. Math. Softw. 29(2), 141–164 (2003). MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Ding, Y.: Dynamic analysis of nonlinear variable frequency water supply system with time delay. Nonlinear Dyn. 90(1), 561–574 (2017). CrossRefzbMATHGoogle Scholar
  7. 7.
    Dutta, G., Doshi, J.B.: A numerical algorithm for the solution of simultaneous nonlinear equations to simulate instability in nuclear reactor and its analysis. In: Computational Science and Its Applications-ICCSA 2011, Lecture Notes in Computer Science, vol. 6783, pp. 695–710. Springer, Berlin (2011).
  8. 8.
    Gupta, S.K., Wahi, P.: Criticality of bifurcation in the tuned axial-torsional rotary drilling model. Nonlinear Dyn. 91(1), 113–130 (2018). CrossRefGoogle Scholar
  9. 9.
    Ishii, M.: Thermally induced flow instabilities in two-phase mixtures in thermal equillibrium. Ph.D. thesis, Georgia Institute of Technology. (1971). Accessed Oct 2017
  10. 10.
    Kakac, S., Bon, B.: A review of two-phase flow dynamic instabilities in tube boiling systems. Int. J. Heat Mass Transf. 51(34), 399–433 (2008). CrossRefzbMATHGoogle Scholar
  11. 11.
    Kuznetsov, Y.: Elements of Applied Bifurcation Theory. Applied Mathematical Sciences. Springer, New York. (2004). Accessed Oct 2017
  12. 12.
    Lee, J., Pan, C.: Dynamics of multiple parallel boiling channel systems with forced flows. Nucl. Eng. Des. 192(1), 31–44 (1999). CrossRefGoogle Scholar
  13. 13.
    Lee, J.D., Lin, Y.G., Chen, S.W., Pan, C.: The influence of void-reactivity feedback on the bifurcation phenomena and nonlinear characteristics of a single nuclear-coupled boiling channel. Ann. Nucl. Energy 94(Supplement C), 814–825 (2016). CrossRefGoogle Scholar
  14. 14.
    Lin, Y., Lee, J., Pan, C.: Nonlinear dynamics of a nuclear-coupled boiling channel with forced flows. Nucl. Eng. Des. 179(1), 31–49 (1998). CrossRefGoogle Scholar
  15. 15.
    Lu, X., Wu, Y., Zhou, L., Tian, W., Su, G., Qiu, S., Zhang, H.: Theoretical investigations on two-phase flow instability in parallel channels under axial non-uniform heating. Ann. Nucl. Energy 63, 75–82 (2014). CrossRefGoogle Scholar
  16. 16.
    Ma, M., Liu, S., Li, J.: Bifurcation of a heroin model with nonlinear incidence rate. Nonlinear Dyn. 88(1), 555–565 (2017). CrossRefzbMATHGoogle Scholar
  17. 17.
    Mishra, A.M., Singh, S.: Subcritical and supercritical bifurcations for two-phase flow in a uniformly heated channel with different inclinations. Int. J. Heat Mass Transf. 93, 235–249 (2016). CrossRefGoogle Scholar
  18. 18.
    Oevelen, T.V., Weibel, J.A., Garimella, S.V.: Predicting two-phase flow distribution and stability in systems with many parallel heated channels. Int. J. Heat Mass Transf. 107, 557–571 (2017). CrossRefGoogle Scholar
  19. 19.
    Paul, S., Singh, S.: A density variant drift flux model for density wave oscillations. Int. J. Heat Mass Transf. 69, 151–163 (2014). CrossRefGoogle Scholar
  20. 20.
    Rizwan-Uddin, Dorning J.: Some nonlinear dynamics of a heated channel. Nucl. Eng. Des. 93(1), 1–14 (1986). CrossRefGoogle Scholar
  21. 21.
    Ruspini, L.C., Marcel, C.P., Clausse, A.: Two-phase flow instabilities: a review. Int. J. Heat Mass Transf. 71, 521–548 (2014). CrossRefGoogle Scholar
  22. 22.
    Saha, P., Ishii, M., Zuber, N.: An experimental investigation of the thermally induced flow oscillations in two-phase systems. ASME J. Heat Transf. 98(4), 616–622 (1976). CrossRefGoogle Scholar
  23. 23.
    Yan, B., Li, R., Wang, L.: The analysis of density wave oscillation in ocean motions with a density variant drift-flux model. Int. J. Heat Mass Transf. 115, 138–147 (2017). CrossRefGoogle Scholar
  24. 24.
    Yan, Y., Zeng, J.: Hopf bifurcation analysis of railway bogie. Nonlinear Dyn. 92, 1–11 (2017). Google Scholar
  25. 25.
    Yun, G., Qiu, S., Su, G., Jia, D.: Theoretical investigations on two-phase flow instability in parallel multichannel system. Ann. Nucl. Energy 35(4), 665–676 (2008). CrossRefGoogle Scholar
  26. 26.
    Zhang, Y., Su, G., Yang, X., Qiu, S.: Theoretical research on two-phase flow instability in parallel channels. Nucl. Eng. Des. 239(7), 1294–1303 (2009). CrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Nuclear Power Corp. of India LtdMumbaiIndia
  2. 2.Indian Institute of Technology-BombayMumbaiIndia

Personalised recommendations