Goodbye to bad vibes: A comparison of HTRI’s tube vibration analysis tools

Andy Mountford, Senior Technical Lead, Training

Most readers are aware that HTRI's shell-and-tube software, Xist^®, includes an integrated analysis of flow-induced tube vibration. Why, then, does HTRI offer a separate tube vibration screening tool, Xvib^®? In this article we review some key differences between the two programs and identify instances in which Xvib is the superlative vibration tool.

What vibration mechanisms does each tool assess?

Xist and Xvib evaluate the risk of tube vibration damage arising from vortex shedding and fluidelastic instability vibration mechanisms.¹ The first step is to accurately determine the fundamental tube natural frequency (f_N).

Tube natural frequency

Xist calculates the fundamental tube natural frequency using the MacDuff-Felgar approach [1], assuming that all spans are of equal length. The program calculates natural frequencies for the longest spans in the inlet, central, and outlet regions, and then uses the lowest of these values in subsequent vibration checks in all regions (Figure 1). In contrast, Xvib uses a 3D finite element method (FEM) discretization to establish mass and stiffness matrices for a single, user-selected tube. The matrices are applied in the undamped equation of motion, from which the eigenvalues (natural frequencies) and eigenvectors (mode shapes) are extracted for the fundamental and higher modes (Figure 2).

Figure 1. Xist approach for determining the fundamental tube natural frequency

Figure 2. Xvib approach for determining the fundamental and higher mode natural frequencies of a selected tube

Velocity profiles and damping

Xist evaluates the vibration potential at 16 locations in the bundle, using area-based velocities adjusted for flow distribution (i.e., stream analysis) and local physical properties (Figure 3).

Figure 3. Xist evaluates tube vibration at 16 locations

Xvib can import an incremental velocity profile and flow orientation from Xist (Figure 4). In Figure 4, the vertical out-of-plane flow orientation in the inlet and outlet regions is orthogonal to the horizontal in-plane flow orientation in the center region. The +/- sign on the incremental velocities denotes opposing flow directions within an orthogonal plane, which the Xist method does not account. The Xist method does, however, account for any perpendicular changes in the flow direction relative to the bundle layout angle, such as those occurring in the inlet/outlet zones compared to the central baffle spaces (Figure 4).

Figure 4. incremental velocity profile and flow orientation input in Xvib

Both Xist and Xvib consider friction, squeeze-film, and viscous damping as a function of the phase of the shellside fluid. Xist calculates a damping contribution, expressed as the log decrement (δ_o) in the inlet, center, and outlet regions, whereas Xvib calculates an average δ_o applied over the entire tube length.

Vortex shedding check

Vortex shedding is a resonance condition characterized by an amplification of the tube vibration amplitude. Xist uses a Strouhal number method to determine the vortex shedding frequency (f_vs) – the frequency at which vortices are shed from the downstream side of the tube. The vibration amplitude is a function of the ratio between the vortex shedding frequency and the tube natural frequency, with maximum displacement occurring at resonance (i.e., f_vs/f_N = 1). Xvib determines the vibration amplitude by solving the damped equation of motion, based on the conservative assumption that the system is at resonance. Tube displacement that exceeds half the tube gap indicates a strong possibility of collision damage.

Fluidelastic instability (FEI) check

Fluidelastic instability refers to a runaway vibrational response when system damping becomes overwhelmed above a critical velocity, V_c . The critical velocity can be predicted using Connors' equation [2]. Xist determines the critical velocity for each region and then uses a basic modal weighting to account for the increased vibration potential in regions with greater modal amplitude. Xvib determines the critical velocity using the tube natural frequency derived from the finite element method and a rigorous modal weighting, comprising an integration of the flow distribution and mode shape. To avoid tube damage, most designers limit the FEI ratio (V /V_c ) to a value less than 0.8 to account for uncertainties.

Summary

Vibration analysis in Xist generally provides a conservative assessment of vibration potential. In some cases, this conservatism can lead to false positive tube vibration warnings. Xvib is commonly used to confirm or dismiss vibration concerns flagged in Xist. If designers are unable to discount spurious vibration concerns, they may implement unnecessary remedial measures that inflate the unit cost.

Xvib can also be deployed to more accurately assess tube vibration in

non-baffled H-shell designs [3]
designs with parallel-cut baffles
designs with variable baffle spacing
designs with annular distributors
designs with internal flow that Xist does not consider [4]
U-tube designs (the Xist analysis can be non-conservative in some cases)

HTRI's training courses include examples of how Xvib can effectively assess the risk of flow-induced vibration damage in such cases. For more information, check out the Events page, or contact our training team at [email protected].

Footnote

¹ Xist also assesses acoustic vibration, which Xvib does not analyze.

Nomenclature

C, Constant with dependency on endzone span to central span ratio and number of spans

E, Modulus of elasticity of tube material, Pa

F_i, Force in the ith row, N

f_N, Natural frequency of fundamental mode, Hz

f_vs, Vortex shedding frequency, Hz

I, Moment of inertia, m⁴

K, Stiffness matrix, N/m or N m/rad

k_i,j, Element of stiffness matrix in the i^th row and j^th column, Nm or N m/rad

L, Length, m

L_i, Distance between tube supports, m

M, Mass matrix, kg

m_i,j, Element of mass matrix in the i^th row and j^th column, kg

t, Time, s

V, Crossflow velocity, m/s

V_c, Connor's critical crossflow velocity for fundamental mode, m/s

W_e, Effective mass per unit length, kg/m

x⃗, Displacement vector, m or rad

y, Weight fraction vapor

δ_o, Log decrement

References

J. N. Macduff and R. P. Felgar, Vibration design charts, Trans. ASME 79, 1459 – 1474 (1957).
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).
Vibration analysis of nonbaffled H shells, TT-26, www.htri.net.
Beware of flow obstructions in inlet and outlet regions, TT-31, www.htri.net.