response to periodic wake shedding must be calculated for any modes that are found to have Strouhal numbers in the range defined in Section 2.4.3. Assuming that the damping is small, peak vibration amplitudes, Y(x), at resonance for the ith mode may be expressed as:
(2‐49)
where FL(x′) is a distributed periodic force formulated by:
where CL is the dynamic lift coefficient defined in Section 2.4.3. Equation (2-50) is a generalized form of Eq. (2-35).
Periodic wake shedding can result from very localized flow over a single span. If the velocity distribution has large step changes in velocity, an effective velocity approach like that used for calculating fluidelastic instability may not be appropriate for periodic wake shedding. Regions with nozzles and inlets may have to be assessed separately.
2.5.4 Acoustic Resonance
Susceptibility to acoustic resonance is not assessed by calculating the tube response. It is estimated by the method described in Sections 2.4.4 and 2.4.5.
2.5.5 Example of Vibration Analysis
An example of a vibration analysis for a typical heat exchanger U‐tube is illustrated in Figs. . This vibration analysis was done with the heat exchanger vibration analysis code PIPO1.
The tube support geometry and the support locations are shown in Fig. 2-20. The input data required for the PIPO1 code is outlined in Fig. 2-21. This figure shows the thermalhydraulic input in the form of pitch flow velocity distribution along the tube. Note the relatively high flow velocity in the inlet region near the tubesheet.
Fig. 2-20 Flow Velocities, Support Locations and Tube Geometry for a Typical Heat Exchanger Tube.
Fig. 2-21 Example of Heat Exchanger Tube Vibration Analysis: Input of Vibration Analysis Code PIPO1.
Typical free vibration analysis results are shown in Fig. 2-22 for selected vibration modes. The results include the vibration mode shapes and the natural frequencies.
Figure 2-23 shows the results of a fluidelastic instability analysis for the condenser tube described in Fig. 2-4. For some vibration modes, the ratio of actual to critical flow velocity for fluidelastic instability, Up/Upc, is greater than one, indicating that fluidelastic instability is possible for this tube, which was subjected to abnormal flow conditions. In reality, fretting‐wear and fatigue damage were observed in this condenser.
Fig. 2-22 Heat Exchanger Tube Vibration: Typical Free Vibration Analysis Results.
2.6 Fretting‐Wear Damage Considerations
Fretting‐wear damage may be assessed using the following methodology, as discussed in detail in Chapter 12. The total fretting‐wear damage over the life of the heat exchanger must not exceed the allowable tube wall reduction or wear depth, dw.
2.6.1 Fretting‐Wear Assessment
Fretting‐wear damage may be estimated using the following modified Archard equation:
(2‐51)
where
where m is the total mass of the tube per unit length, and
Fig. 2-23 Vibration Mode Shapes and Vibration Analysis Results for a Typical Condenser Tube.
2.6.2
