Andy Mountford, Senior Technical Lead, Training
Peter Joosten, Manager, Technical Support

In a previous article, we reviewed the differences between the Xist® and Xvib® vibration analysis methods for straight-tube heat exchanger configurations. These differences are summarized in the table below.

Straight-tube method Xist Xvib
Vibration mechanisms analyzed Fluidelastic instability, vortex shedding, and acoustic vibration Fluidelastic instability and vortex shedding
Bundle locations analyzed 16 locations: shell/bundle entrance/exit regions, as well as inlet, center, and outlet regions User selects individual tubes from any location in bundle
Support geometries and flow configurations accommodated Velocities and tube support geometry defined by Xist geometry inputs; uniform flow orientation along tube Crossflow velocity profile, flow orientation, and tube support configuration defined by user
Tube natural frequency calculation MacDuff-Felgar method; uses lowest straight span natural frequency of inlet, center, and outlet; 1st mode only; considers axial loading if specified Finite element method (FEM) for natural frequencies; 1st – 15th mode, plus mode shapes; axial loading not considered
Vortex shedding check fvs from Strouhal numbers; vibration amplitude based on fvs/fN, magnification factor and damping Vibration amplitude from damped equation of motion assuming resonance condition
Fluidelastic instability check Regional velocities compared to Connors' critical velocity adjusted by simplified modal weighting Modal weighted spanwise velocities compared to Connors' critical velocity
Damping Log decrement calculated in each region (inlet, center, outlet) Overall average log decrement
Impingement plate jetting analysis Not available Scaled Xist nozzle velocity applied to tubes imported from plate edges [1]

In this article, we review the differences between the Xist and Xvib vibration analysis methods for heat exchanger designs containing U-tube bundles.

U-tube bundles

Compared to straight tubes of equal span length, U-tubes are more prone to flow-induced vibration and its harmful effects. Theoretically, U-tubes can exhibit two modes of tube vibration: in-plane and out-of-plane. The in-plane mode distorts the radius of the bend locally, while the out-of-plane mode is the first bending mode for a cantilevered beam (Figure 1).

Figure 1. In-plane and out-of-plane vibration responses of U-bends
Figure 1. In-plane and out-of-plane vibration responses of U-bends

Out-of-plane modes are more serious, because they occur at lower frequencies—roughly 80% of the natural frequency of a straight tube of equal span length [2]. In contrast, in-plane modes have never been observed to be unstable, even though the crossflow velocity may be significantly greater than the calculated critical velocity [3]. For the U-bend portion of the tube, Xist uses the method from the TEMA Standards to calculate the natural frequency [4].

calculate u-bend natural frequency

The stable in-plane mode is excited by an out-of-plane force, while the potentially unstable out-of-plane mode is excited by an in-plane force, as shown in Figure 2.

Figure 2. Vibration response is perpendicular to flow direction

Figure 2. Vibration response is perpendicular to flow direction

Although Xist calculates the U-bend natural frequency according to the equation above, the program uses the lowest straight span natural frequency in the subsequent vibration checks.1 For this reason, Xist issues a runtime warning for designs in which the longest unsupported span occurs in the U bend region because the Xist vibration analysis of these cases may be non-conservative. Most designers generally avoid designs in which the longest unsupported span occurs in the U-bend region, even when a full support plate is located at the U-bend tangent, because flow excitation in a straight section of the tube can propagate along the tube structure and provoke a vibration response in the U-bend.

The Xvib finite element method, described in an earlier article, provides a more accurate calculation of the fundamental and higher mode natural frequencies of U-tubes. This method more precisely evaluates the vibration potential across nearly all U-tube configurations, including those in which the longest unsupported span is located in the U-bend, as detailed in an upcoming case study.

Footnote

1 However, when a perpendicular baffle cut orientation and nozzle position are set at (i.e., above) the U-bends, the calculated U-bend natural frequency is used for the shell/bundle entrance/exit analysis.

Nomenclature

Cu, Mode constant of u-bend

Cuws, Multiplier based on number of intermediate tube supports in U-bend

E, Modulus of elasticity of tube material, Pa

fN, Natural frequency of fundamental mode, Hz

fu, U-bend natural frequency, Hz

fvs, Vortex shedding frequency, Hz

I, Moment of inertia, m4

Rb, Mean bend radius, m

We, Effective mass per unit length, kg/m

References

  1. J. N. Macduff and R. P. Felgar, Vibration design charts, Trans. ASME 79, 1459 – 1474 (1957).
  2. H. J. Connors, Fluidelastic vibration of tube arrays excited by nonuniform cross flow, in Proc. Pressure Vessels Piping Conf., ed. M. K. Au-Yang, 93 – 107, ASME, New York (1980).
  3. Vibration analysis of nonbaffled H shells, TT-26, www.htri.net.
  4. Beware of flow obstructions in inlet and outlet regions, TT-31, www.htri.net.