various methods of normalizing their results. Therefore, it was necessary to select one means of normalization and apply it to all of the data. The adopted method is the “equivalent power spectral density (EPSD),” first described by Axisa et al (1990).
Fig. 2-13 Effect of P/D on Fluidelastic Instability Constant in Two‐Phase Cross Flow.
The power spectral density (PSD),
(2‐27)
where, fR is the reduced frequency, defined as f/fo, and D is the tube diameter.
A difficulty arises in the calculation of SF(f) because the correlation length, λc, is rarely known. Axisa et al (1990) present a dimensionless EPSD,
(2‐28)
where, Le 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,
where, ϕ1(x1) is the normalized mode shape for the 1st mode, a1 is the numerical coefficient for the 1st mode, f1 is the 1st mode tube natural frequency, m is the total tube mass (tube mass + hydrodynamic mass) and ζ1 is the damping ratio for the 1st mode. Values of
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,
where, Le, is the excited tube length. In this chapter, reference lengths of Lo = 1 m and Do = 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:
Fig. 2-14 Proposed Guideline for Single‐Phase Random Excitation Forces (References Provided in the Legend can be found in Taylor and Pettigrew (2000)).
(2.31)
(2‐32)
For single‐phase flow fo = Up/D and
In most cases, the random excitation forces for interior tubes are significantly lower than for upstream tubes. The vibration response of the upstream tubes will be larger. Thus, it may not be necessary to consider the vibration response of interior tubes when they are otherwise identical to the upstream tubes.
Two‐Phase Cross Flow
Random turbulence excitation forces can be significant in two‐phase cross flow, in particular, in the U‐bend region of steam generators. The term turbulence is used loosely in two‐phase flows. It describes the dynamics of the two‐phase mixture as it flows through a tube bundle.
For void fractions of 10% or less, two‐phase flow random forces behave like single‐phase flow forces and the single‐phase guidelines can be used. At higher void fractions, the effect of the two‐phase mixture begins to dominate the single‐phase random forces. The physics of these two‐phase forces are not well understood but recent study of the effect of void fraction and flow regime have led to reasonable collapse of two‐phase data from air‐water, Freon liquid‐vapor, Freon vapor‐water and steam‐water experiments.
Preliminary two‐phase design guidelines for random excitation of heat exchanger tubes were presented by several authors such as Taylor et al (1989),
