Flow-Induced Vibration Handbook for Nuclear and Process Equipment. Группа авторов

Fig. 2-6 Hydrodynamic Mass in Two‐Phase Cross Flow: Comparison Between Theory and Experiments. (Note that m_ℓ is the hydrodynamic mass per unit length in liquid.)

(2.9)

      where D_e is the equivalent diameter of the surrounding tubes and the ratio D_e/D is a measure of confinement. The effect of confinement is formulated by

      (2‐10)

      for a tube inside a triangular tube bundle (Rogers et al, 1984). Similarly, for a square tube bundle (Pettigrew et al, 1989a) confinement may be approximated by

      (2‐11)

The hydrodynamic mass of tube bundles subjected to two‐phase cross flow may be calculated with Eq. (2-9) provided that the homogeneous density of the two‐phase mixture, ρ_TP, is used in the formulation (Pettigrew et al, 1989a). Figure 2-6 compares Eq. (2-9) to experimental data.

      The total dynamic mass of the tube per unit length, m, comprises the hydrodynamic mass per unit length, m_h, the tube mass per unit length, m_t, and the mass per unit length of the fluid inside the tube, m_i:

      (2‐12)

      See Chapter 4 for more detail on hydrodynamic mass.

      2.3.2 Damping

      Vibration energy dissipation (damping) is an important parameter in limiting vibration. Damping in single‐ and two‐phase flows is discussed in detail in Chapters 5 and 6, respectively.

      Heat Exchanger Tubes in Gases

      As discussed in Pettigrew et al (1986), the dominant damping mechanism in heat exchangers with gas on the shell‐side is friction between tubes and tube supports. The available information on damping of heat exchanger tubes in gases has been reviewed. This work yielded the following design recommendation for estimating the friction damping ratio, ζ_F, in percent:

(2‐13)

      which takes into account the effect of support thickness, L, span length, ℓ_m, and number of spans, N.

      Heat Exchanger Tubes in Liquids

      As discussed in Chapter 5, there are three important energy dissipation mechanisms that contribute to damping of multi‐span heat exchanger tubes with liquids on the shell‐side. These are viscous damping between tube and liquid, squeeze‐film damping in the clearance between tube and tube support, and friction damping at the support. Thus, the total tube damping, ζ_T, which we define as the ratio of actual to critical damping in percent, is expressed by

(2‐14)

      where, ζ_V, ζ_SF and ζ_F are, respectively, the viscous, squeeze‐film and friction damping ratios.

      Tube‐to‐fluid viscous damping is related to the Stokes number, πfD²/2v, and the degree of confinement of the heat exchanger tube within the tube bundle or D/D_e. Rogers et al (1984) developed a simplified formulation for viscous damping, valid for πfD²/2v > 3300 and D/D_e < 0.5 which covers most heat exchangers. Their simplified expression for ζ_V(in percent) is

(2‐15)

where v is the kinematic viscosity of the fluid and f is the natural frequency of the tube for the mode being analysed. Clearly, viscous damping is frequency dependent. Calculated values of damping using Eq. (2-15) are compared against experimental data in Fig. 2-7. The comparison shows reasonable agreement.

Squeeze‐film damping, ζ_SF, and friction damping, ζ_F, take place at the supports. Semi‐empirical expressions were developed, based on available experimental data, to formulate friction and squeeze‐film damping as discussed by Pettigrew et al (2011) and in Chapter 5. The available experimental data is outlined in Fig. 2-8. Thus, squeeze‐film damping (in percent) may be formulated by

(2‐16)

      and friction damping (in percent) by

(2‐17)

where (N − 1)/N takes into account the ratio of the number of supports over the number of spans, L is the support thickness and ℓ_m is a characteristic tube length. The latter is defined as the average of the three longest spans when the lowest modes and the longest spans dominate the vibration response. This is usually the case. When higher modes and shorter spans govern the vibration response, then, the characteristic span length should be based on these shorter spans. This could happen when there are high flow velocities locally,