Advertisement

Effect of nonlinear cladding stiffness on the stability and Hopf bifurcation of a heat-exchanger tube subject to cross-flow

  • 16 Accesses

Abstract

The linear stability of a heat-exchanger tube modeled as a single-span cantilever beam subjected to cross-flow has been studied with two parameters: (1) varying stiffness of the baffle-cladding at the free end and (2) varying flow velocity. A mathematical model incorporating the motion-dependent fluid forces acting on the beam is developed using the Euler–Bernoulli beam theory, under the inextensible condition. The partial delay differential equation governing the dynamics of the continuous system is discretized to a set of finite, nonlinear delay differential equations through a Galerkin method in which a single mode is considered. Unstable regions in the parametric space of dimensionless cladding stiffness and flow velocity are identified, along with the magnitude of damping in the stable region. This information can be used to determine the cladding stiffness at which the system should be operated to achieve maximum damping at a known operational flow velocity. Furthermore, the system is found to lose stability by Hopf bifurcation and the method of multiple scales is used to analyze its post-instability behavior. Stable and unstable limit cycles are observed for different values of the linear component of the dimensionless cladding stiffness. A global bifurcation analysis indicates that the number of limit cycles decreases with increasing linear cladding stiffness. An optimal range for the linear cladding stiffness is recommended where tube vibrations would either diminish to zero or assume a relatively low amplitude associated with a stable limit cycle.

This is a preview of subscription content, log in to check access.

Access options

Buy single article

Instant unlimited access to the full article PDF.

US$ 39.95

Price includes VAT for USA

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13

Abbreviations

L :

Tube length (m)

\(\zeta\) :

Damping ratio

\(\rho\) :

Fluid density (kg m\(^{-3}\))

D :

Outer diameter of tube (m)

t :

Dimensionless time

X :

Axial spatial coordinate (m)

T :

Time (s)

\(\mu\) :

Flow retardation parameter

\({\mathcal {O}}\) :

Order of magnitude

x :

Dimensionless axial spatial coordinate

w :

Dimensionless transverse displacement of tube

\(\tilde{f}\delta (X-X_b)\) :

Restraining force due to spring at \(X=X_b\) (N m\(^{-1}\))

A :

Cross-sectional area of cylinder (\(\hbox {m}^2\))

W :

Transverse displacement of tube (m)

\(\tilde{U}\) :

Cross-flow velocity (m s\(^{-1}\))

\(f\delta (x-x_b)\) :

Dimensionless spring force

E :

Young’s modulus of tube material (Pa)

\(C_{D}\) :

Drag coefficient

\(C_{L}\) :

Lift coefficient

M :

Mass per unit length of tube (kg m\(^{-1}\))

C :

Viscous damping coefficient (N s m\(^{-2}\))

I :

Second moment of inertia of tube cross-section (m\(^{4}\))

\(\delta _1\) :

Modal logarithmic decrement

\(C_{ma}\) :

Added-mass coefficient

\(\varDelta T\) :

Time delay (s) \(\left( \frac{\mu D}{ \tilde{U}}\right)\)

m :

Tube mass-parameter \(\left( \frac{M}{\rho D^{2}}\right)\)

U :

Dimensionless flow-velocity \(\left( \frac{2\pi \tilde{U}}{D\varOmega _1}\right)\)

\(\omega _{cr}\) :

Dimensionless critical angular frequency

\(\tau\) :

Dimensionless time delay due to flow retardation \(\left( \frac{2\pi }{U}\right)\)

q :

Dimensionless tube vibration response

\(\varOmega _1\) :

Frequency of first mode (\(\hbox {s}^{-1}\))

\(\lambda _1\) :

Dimensionless eigenvalue of first mode

F :

Cross-flow-induced force per unit length (N m\(^{-1}\))

\(K_1\) :

Linear spring stiffness (N m\(^{-1}\))

\(K_2\) :

Cubic spring stiffness (N m\(^{-3}\))

\(k_1\) :

Dimensionless linear spring stiffness \(\left( \frac{K_1L^4}{\lambda _1^4EID}\right)\)

\(k_1^{cr}\) :

Dimensionless critical linear spring stiffness

\(k_2\) :

Dimensionless cubic spring stiffness \(\left( \frac{K_2L^6}{\lambda _1^4EID}\right)\)

P :

Pitch (m)

References

  1. 1.

    Paıdoussis MP, Li GX (1992) Cross-flow-induced chaotic vibrations of heat-exchanger tubes impacting on loose supports. J Sound Vib 152(2):305–326

  2. 2.

    Chen SS (1983) Instability mechanisms and stability criteria of a group of circular cylinders subjected to cross-flow. Part I: theory. J Vib Acoust Stress Reliab Des 105(1):51–58

  3. 3.

    Lever J, Weaver D (1982) A theoretical model for fluid-elastic instability in heat exchanger tube bundles. J Press Vessel Technol 104(3):147–158

  4. 4.

    Khalifa A, Weaver D, Ziada S (2012) A single flexible tube in a rigid array as a model for fluidelastic instability in tube bundles. J Fluids Struct 34:14–32

  5. 5.

    Wang L, Ni Q (2010) Hopf bifurcation and chaotic motions of a tubular cantilever subject to cross flow and loose support. Nonlinear Dyn 59(1–2):329–338

  6. 6.

    Xia W, Wang L (2010) The effect of axial extension on the fluidelastic vibration of an array of cylinders in cross-flow. Nucl Eng Des 240(7):1707–1713

  7. 7.

    Wang L, Dai HL, Han YY (2012) Cross-flow-induced instability and nonlinear dynamics of cylinder arrays with consideration of initial axial load. Nonlinear Dyn 67(2):1043–1051

  8. 8.

    Sadath A, Dixit HN, Vyasarayani CP (2016) Dynamics of cross-flow heat exchanger tubes with multiple loose supports. J Press Vessel Technol 138(5):051303

  9. 9.

    Sadath A, Vinu V, Vyasarayani CP (2017) Vibrations of a simply supported cross flow heat exchanger tube with axial load and loose supports. J Comput Nonlinear Dyn 12(5):051001

  10. 10.

    Cai Y, Chen S (1993) Chaotic vibrations of nonlinearly supported tubes in crossflow. J Press Vessel Technol 115(2):128–134

  11. 11.

    Weaver D, Grover L (1978) Cross-flow induced vibrations in a tube bank-turbulent buffeting and fluid elastic instability. J Sound Vib 59(2):277–294

  12. 12.

    Weaver D, El-Kashlan M (1981) The effect of damping and mass ratio on the stability of a tube bank. J Sound Vib 76(2):283–294

  13. 13.

    Price S, Paıdoussis M (1984) An improved mathematical model for the stability of cylinder rows subject to cross-flow. J Sound Vib 97(4):615–640

  14. 14.

    Li H, Mureithi N (2017) Development of a time delay formulation for fluid-elastic instability model. J Fluids Struct 70:346–359

  15. 15.

    Mureithi NW, Paidoussis MP, Price SJ (1994) The post-hopf-bifurcation response of a loosely supported cylinder in an array subjected to cross-flow. Part II: theoretical model and comparison with experiments. J Fluids Struct 8(7):853–876

  16. 16.

    de Pedro B, Parrondo J, Meskell C, Oro JF (2016) CFD modelling of the cross-flow through normal triangular tube arrays with one tube undergoing forced vibrations or fluid-elastic instability. J Fluids Struct 64:67–86

  17. 17.

    de Pedro Palomar B, Meskell C (2018) Sensitivity of the damping controlled fluidelastic instability threshold to mass ratio, pitch ratio and Reynolds number in normal triangular arrays. Nucl Eng Des 331:32–40

  18. 18.

    Sawadogo T, Mureithi N (2014) Fluidelastic instability study on a rotated triangular tube array subject to two-phase cross-flow. Part II: experimental tests and comparison with theoretical results. J Fluids Struct 49:16–28

  19. 19.

    Piteau P, Delaune X, Borsoi L, Antunes J (2019) Experimental identification of the fluid-elastic coupling forces on a flexible tube within a rigid square bundle subjected to single-phase cross-flow. J Fluids Struct 86:156–169

  20. 20.

    Mahon J, Meskell C (2013) Estimation of the time delay associated with damping controlled fluidelastic instability in a normal triangular tube array. J Press Vessel Technol 135(3):030903

  21. 21.

    Meskell C (2009) A new model for damping controlled fluidelastic instability in heat exchanger tube arrays. Proc Inst Mech Eng A J Power Energy 223(4):361–368

  22. 22.

    El Bouzidi S, Hassan M (2015) An investigation of time lag causing fluidelastic instability in tube arrays. J Fluids Struct 57:64–276

  23. 23.

    Lever JH, Weaver DS (1986) On the stability of heat exchanger tube bundles, part I: modified theoretical model. J Sound Vib 107(3):375–392

  24. 24.

    Wang L, Jiang T, Dai H, Ni Q (2018) Three-dimensional vortex-induced vibrations of supported pipes conveying fluid based on wake oscillator models. J Sound Vib 422:590–612

  25. 25.

    Liu Z, Wang L, Dai H, Wu P, Jiang T (2019) Nonplanar vortex-induced vibrations of cantilevered pipes conveying fluid subjected to loose constraints. Ocean Eng 178:1–19

  26. 26.

    Duan J, Chen K, You Y, Li J (2018) Numerical investigation of vortex-induced vibration of a riser with internal flow. Appl Ocean Res 72:110–121

  27. 27.

    Jiang T, Liu Z, Dai H, Wang L, He F (2019) Nonplanar multi-modal vibrations of fluid-conveying risers under shear cross flows. Appl Ocean Res 88:187–209

  28. 28.

    Yuan Y, Xue H, Tang W (2018) Numerical analysis of vortex-induced vibration for flexible risers under steady and oscillatory flows. Ocean Eng 148:548–562

  29. 29.

    Wahi P, Chatterjee A (2005) Galerkin projections for delay differential equations. J Dyn Syst Meas Control 127(1):80–87

  30. 30.

    Vyasarayani C, Subhash S, Kalmár-Nagy T (2014) Spectral approximations for characteristic roots of delay differential equations. Int J Dyn Control 2(2):126–132

  31. 31.

    He JH (2001) Bookkeeping parameter in perturbation methods. Int J Nonlinear Sci Numer Simul 2(3):257–264

  32. 32.

    Das S, Chatterjee A (2002) Multiple scales without center manifold reductions for delay differential equations near Hopf bifurcations. Nonlinear Dyn 30(4):323–335

  33. 33.

    Nayfeh AH, Balachandran B (2008) Applied nonlinear dynamics: analytical, computational, and experimental methods. Wiley, New York

  34. 34.

    Pettigrew MJ, Yetisir M, Fisher NJ, Smith BA, Taylor CE, Janzen VP (2017) Fretting-wear damage due to vibration in nuclear and process equipment. In: ASME 2017 pressure vessels and piping conference, p V004T04A032

Download references

Acknowledgements

The authors gratefully acknowledge the Department of Science and Technology for funding this research through the Inspire Fellowship (DST/INSPIRE/04/2014/000972).

Author information

Correspondence to Ajinkya Desai.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Vourganti, V., Desai, A., Samukham, S. et al. Effect of nonlinear cladding stiffness on the stability and Hopf bifurcation of a heat-exchanger tube subject to cross-flow. Meccanica (2020) doi:10.1007/s11012-019-01114-z

Download citation

Keywords

  • Heat-exchanger tube
  • Linear stability
  • Nonlinear cladding stiffness
  • Delay differential equation
  • Method of multiple scales
  • Hopf bifurcation