# A note on damping in heat-exchanger tubes subjected to cross-flow

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## Abstract

The equation governing the dynamics of a heat-exchanger tube is a delay differential equation (DDE). In all the earlier studies, only the stability boundaries in the parametric space of mass-damping parameter and reduced flow-velocity were reported. The contour plots showing the damping in different regions of the stability chart has never been reported, due to the complexity in solving the infinite-dimensional nonlinear eigenvalue problem associated with characteristic roots of the governing DDE. In this work using Galerkin approximations, the spectrum (characteristic roots) of the DDE is obtained. The rightmost characteristic root, whose real part represents the damping in the heat-exchanger tube is included in the stability chart. Interestingly, it is found that the highest damping is present in localized areas of the stability charts, which are close to the stability boundaries. These stability charts can be used to determine the optimal cross-flow velocities for operating the heat-exchanger tube for achieving maximum damping. Explicit evaluation of the characteristic roots allows us to show that the roots cross the stability boundary with a non-zero velocity, clearly indicating the existence of Hopf bifurcation at the stability boundary.

## Keywords

Heat-exchanger tube Delay differential equation Nonlinear eigenvalue problem Stability chart Damping contours## List of symbols

- 2
*T* Distance between centres of two adjacent cylinders of a row

- \(\bar{\beta }\)
Fluid damping term for cylinder motion in lift direction

- \(\bar{\kappa }\)
Fluid stiffness term for cylinder motion in lift direction

- \(\bar{\lambda }\)
Dimensionless eigenvalue of a matrix

- \(\bar{m}\)
Mass-parameter of a tube \(\left( \frac{m}{\rho d^{2}}\right) \)

- \(\delta \)
Modal logarithmic decrement

- \(\lambda \)
Dimensionless roots of characteristic equation

- \(\mathcal {O}\)
Order of magnitude

- \(\mu \)
Flow retardation parameter

- \(\omega _{n}\)
Natural angular frequency of tube

- \(\rho \)
Fluid density

- \(\tau \)
Dimensionless time-delay due to flow retardation \(\left( \frac{2\pi \mu }{aU_{r}}\right) \)

*c*Modal viscous damping coefficient for cylindrical tube

- \(C_{D0}\)
Drag coefficient for stationary cylinder

- \(C_{L}\)
Lift coefficient for flexible cylinder

*d*Tube diameter

*e*Modal stiffness coefficient for cylindrical tube

*f*Natural frequency of tube \(\left( \frac{\omega _{n}}{2\pi }\right) \)

*L*Streamwise distance between consecutive cylinder rows

*l*Length of cylindrical tube

*m*Mass per unit length of cylindrical tube

*P*Pitch \((2T-d)\)

*t*Dimensionless time

- \(U_{\infty }\)
Free-stream velocity

- \(U_{c}\)
Free-stream velocity at onset of transverse instability, also called critical velocity

- \(U_{G}\)
Velocity of flow downstream of first row of upstream cylinders, also called gap velocity

- \(U_{rc}\)
Reduced critical velocity \(\left( \frac{U_{c}}{fd}\right) \)

- \(U_{r}\)
Reduced flow-velocity \(\left( \frac{U_{\infty }}{fd}\right) \)

*y*Nondimensional transverse displacement of flexible tube

## Notes

### Acknowledgements

The authors gratefully acknowledge the anonymous reviewers for reviewing this paper.

### Funding

Funding was provided to C.P. Vyasarayani by the Department of Science and Technology through the Inspire fellowship (Grant Number DST/INSPIRE/04/2014/000972). The funders had no role in designing the study, collecting or analyzing data, the decision to publish, or preparing the manuscript

### Compliance with ethical standards

### Conflict of interest

The authors have no competing interests to declare.

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