Fig. 2-9 Comparison Between Tube Support Damping Model (Squeeze‐Film and Friction) and Experimental Data.

Equations (2-16) and (2-17) are combined to determine the squeeze‐film damping parameter, 1460/f, and compared against available experimental data in Fig. 2-9. There is large scatter in the data, which is mostly due to the nature of the problem. Heat exchanger tube damping depends on parameters that are difficult to control, such as support alignment, tube straightness, relative position of tube within the support, and side loads. These parameters are statistical in nature and probably contribute to much of the scatter. A conservative but realistic criterion for damping is obtained by taking damping values at roughly the lower decile of the available data, as shown in Fig. 2-8.

Substituting Eqs. (2-15), (2-16) and (2-17) in Eq. (2-14):

(2‐18)

Although somewhat speculative, Eq. (2-18) formulates all important energy dissipation mechanisms and fits the data best. Thus, it is our recommendation as a damping criterion for design purposes.

However, if the damping ratio predicted by this equation is less than 0.6%, we recommend taking a minimum value of 0.6%. As shown in Fig. 2-8, a minimum damping of 0.6% appears reasonable. In Fig. 2-8, there is no experimental data for frequencies below 25 Hz. Rather than extrapolating Eq. 2-18 to lower frequencies, we recommend using a maximum total damping of 5%.

      Damping in Two‐Phase Flow

      The subject of heat exchanger tube damping in two‐phase flow was reviewed by Pettigrew and Taylor (1997) as discussed in Chapter 6. The total damping ratio, ζ_T, of a multi‐span heat exchanger tube in two‐phase flow comprises support damping, ζ_S, viscous damping, ζ_V, and two‐phase damping, ζ_TP:

      (2‐19)

Depending on the thermalhydraulic conditions (i.e., heat flux, void fraction, flow, etc.), the supports may be dry or wet. If the supports are dry, which is more likely for high heat flux and very high void fraction, only friction damping takes place. Support damping in this case is analogous to damping of heat exchanger tubes in gases, Pettigrew et al (1986). Damping due to friction (and impact) in dry supports may be expressed by Eq. (2-13). The above situation is unlikely in well‐designed recirculating steam generators. However, it could exist in some areas of once‐through steam generators.

      When there is liquid between the tube and the support, support damping includes both squeeze‐film damping, ζ_SF, and friction damping, ζ_F. This situation is analogous to heat exchanger tubes in liquids and the support damping may be evaluated with Eqs. (2.18) and (2.19) above:

(2‐20)

Note that in Eq. (2-20), ρ_ℓ is the density of the liquid within the tube support, whereas m is the total mass per unit length of the tube including the hydrodynamic mass calculated with the two‐phase homogeneous density.

      Viscous damping in two‐phase mixtures is analogous to viscous damping in single‐phase fluids (Pettigrew and Taylor, 1997). Homogeneous properties of the two‐phase mixture are used in its formulation, as follows:

      (2‐21)

      where v_TP is the equivalent two‐phase kinematic viscosity as per McAdams et al (1942):

      (2‐22)

      Above 40% void fraction, viscous damping is generally small and could be neglected for the U‐bend region of steam generators. However, it is significant for lower void fractions.

      There is a two‐phase component of damping in addition to viscous damping. As discussed by Pettigrew and Taylor (1997) and in Chapter 6, two‐phase damping is strongly dependent on void fraction, fluid properties and flow regimes, directly related to confinement and the ratio of hydrodynamic mass over tube mass, and weakly related to frequency, mass flux or flow velocity, and tube bundle configuration. A semi‐empirical expression was developed from the available experimental data to formulate the two‐phase component of damping, ζ_TP, in percent:

      (2‐23)

Fig. 2-10 Comparison Between Proposed Design Guideline and Available Damping Data. Normalized Two‐Phase Damping Ratio (ζ_TP)_n:

As shown in Fig. 2-10, the void‐fraction function, f(ε_g), may be approximated by taking the envelope through the lower decile of the data:

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