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

Fig. 2-13 Effect of P/D on Fluidelastic Instability Constant in Two‐Phase Cross Flow.

The power spectral density (PSD), , can be rendered dimensionless using a pressure scaling factor, p_o, and a frequency scaling factor, f_o, as follows:

      (2‐27)

      where, f_R is the reduced frequency, defined as f/f_o, and D is the tube diameter.

      A difficulty arises in the calculation of S_F(f) because the correlation length, λ_c, is rarely known. Axisa et al (1990) present a dimensionless EPSD, , defined as follows:

      (2‐28)

      where, L_e is the excited tube length. Using this definition, the dimensionless EPSD for Mode 1 can be defined in terms of the mean square of tube displacement, as follows:

(2‐29)

      where, ϕ₁(x₁) is the normalized mode shape for the 1^st mode, a₁ is the numerical coefficient for the 1^st mode, f₁ is the 1^st mode tube natural frequency, m is the total tube mass (tube mass + hydrodynamic mass) and ζ₁ is the damping ratio for the 1^st mode. Values of and a₁ are 2.0 and 1.1, respectively, for pinned‐pinned end conditions.

Using Eq. (2-29), the mean square of tube displacement can be found without knowledge of the correlation length. Instead, a small correlation length is assumed. To correctly compare spectra obtained using experimental rigs with varying geometries, it is necessary to define a dimensionless reference EPSD, , based on a reference excited tube length, L_o, and a reference tube diameter, D_o, as follows:

(2‐30)

      where, L_e, is the excited tube length. In this chapter, reference lengths of L_o = 1 m and D_o = 0.02 m are applied.

      Single‐Phase Cross Flow

      In single‐phase cross flow, two distinct flow fields are possible. Interior tubes, well within a heat exchanger tube bundle, are excited by turbulence generated within the bundle. This excitation is governed by the tube bundle geometry. On the other hand, upstream or inlet tubes are excited by turbulence generated by upstream components such as inlet nozzles, entrance ports and upstream piping elements. Upstream turbulence levels are governed by the upstream flow path geometry and are very often much larger than those generated within the bundle. Such excitation is often referred to as far‐field excitation.

      Random turbulence excitation is usually not a problem with gas or vapor cross flow. The pressure fluctuations and resulting excitation forces due to gas cross flow at a given velocity are generally an order of magnitude less than those for a liquid or two‐phase mixture at the same velocity. However, gas velocities can be extremely high and at high pressure the densities can be significant. Therefore, some consideration should be given to random excitation in high‐pressure gas heat exchangers.

Taylor and Pettigrew (2000) combined data from many sources to arrive at the reference EPSD guideline shown in Fig. 2-14. The lower bound, shown in Fig. 2-14, should be used when the upstream turbulence is less than or equal to the turbulence within the tube bundle. The upper bound, shown in Fig. 2-14, should be used if the upstream turbulence exceeds the turbulence inside the tube bundle. The boundaries are defined as follows: