Читать книгу Flow-Induced Vibration Handbook for Nuclear and Process Equipment - Группа авторов - Страница 34

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:

(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 ζ_S = 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 and are, therefore, contacting or vibrating very close to one side of the supports. It is difficult to imagine that many tubes of typically one‐metre span length, would be located concentrically within a 0.38 mm diametral clearance without touching the support. The chances of a tube not touching a support are probably much less than one percent. Thus, it is reasonable to assume pinned conditions at the supports to allow for a quasi‐linear vibration analysis. For the purpose of vibration analysis, drilled hole, broached hole, and egg‐crate type supports may be taken as pinned supports.

Anti‐vibration bars (AVBs) are often used in the U‐bend region of nuclear steam generators. AVBs are reasonably effective in the out‐of‐plane direction, but offer little restraint in the in‐plane (plane of the U‐bend) direction (Taylor et al, 1995). Fortunately, the vibration response is usually much less in the in‐plane direction than in the out‐of‐plane direction. Nevertheless, AVBs are less effective than other clearance supports. For steam generator vibration analyses, it is recommended that one (any one) AVB in the U‐bend region be assumed ineffective. The possibility that two adjacent AVBs would be ineffective at the same time is considered too improbable to be of concern. For example, if we assume that the probability of one AVB being ineffective is less than one percent, then the probability of two adjacent AVBs being ineffective is less than 0.01% or less than one tube in 10 000, which is less than one tube per steam generator and, thus, insignificant.

In the in‐plane direction, on the other hand, recent experience has shown that more than one adjacent AVB may be ineffective and that maybe two or more AVBs should be assumed ineffective in the in‐plane direction. In practice, tube‐to‐support clearances must be small enough to provide an effective support. Thus, pinned support conditions may be assumed provided that the tube‐to‐support diametral clearance for drilled holes, broached holes, scallop bars, egg crates and lattice bars is equal to or less than 0.4 mm. In the out‐of‐plane direction, the diametral clearance between tube and AVBs should be sufficiently small to provide effective support (e.g., <0.1 mm).