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):
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, ζT, of a multi‐span heat exchanger tube in two‐phase flow comprises support damping, ζS, viscous damping, ζV, and two‐phase damping, ζTP:
(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, ζSF, and friction damping, ζF. 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:
Note that in Eq. (2-20), ρℓ 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 vTP 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, ζTP, in percent:
(2‐23)
Fig. 2-10 Comparison Between Proposed Design Guideline and Available Damping Data. Normalized Two‐Phase Damping Ratio (ζTP)n:
As shown in Fig. 2-10, the void‐fraction function, f(εg), may be approximated by taking the envelope through the lower decile of the data:
(2‐24)
The fluid properties/flow regime dependence was difficult to assess in the absence of a broad range of damping data for different two‐phase mixtures. Flow regime effects are partly taken care of by the void‐fraction function.
Tube fouling is not expected to contribute much to damping. However, support damping is considerably reduced by crudding within the support. At the limit, when tubes are jammed in the support by severe crud deposition, a support damping value of ζS = 0.2% should be used in analysis.
Dynamic Stiffness and Support Effectiveness The dynamic stiffness of multi‐span heat exchanger tubes is simply the flexural rigidity, EI, where E is the elastic modulus of the tube material and I is the area moment of inertia of the tube cross‐section. The boundary conditions, that is, the support conditions, are somewhat complex. The tubes are effectively clamped at the tubesheet. To facilitate assembly and to allow for thermal expansion, there is a clearance between tubes and tube supports. The diametral clearance between tube and intermediate support is typically 0.25 to 0.80 mm (for many nuclear heat exchangers, the diametral clearance is specified to be 0.38 mm or 0.015 in.). Thus, the dynamic interaction between tube and tube support is inherently non‐linear. In well‐designed heat exchangers, the tube vibration response at mid‐span is mostly less than 100 μm root‐mean‐square (rms) and much less at the supports (usually less than 25 μm rms). This response is significantly less than the available diametral clearance. Thus, the tubes do not generally vibrate back and forth across the available clearance. Instead, most tubes are not centered within the supports
