Added mass and damping are known to be dependent on fluid properties (in particular, fluid density and viscosity) as well as functions of component geometry and adjacent boundaries, whether rigid or elastic. Nuclear reactor components are typically immersed in a liquid coolant and are often closely spaced (Wambsganss, et al., 1974).

3.1 Added mass

(TEMA, 7th Edition, 1988) defines hydrodynamic mass as an effect which increases the apparent weight of the vibrating body due to the displacement of the shell-side flow resulting from motion of vibrating tubes, the proximity of other tubes within the bundles and relative location of shell-wall. The so-called "virtual mass" for a tube is composed of the mass of the tube, mass of the fluid contained in the tube and the inertia M’ imposed by the surrounding fluid. This hydrodynamic mass M’ is a function of the geometry, the density of the fluid, and the size of the tube. In an ideal fluid, it is proportional to the fluid density and to the volume of the tube (Moretti & Lowry, 1976), and hence may be expressed, per unit length as:

M’ C 2 (-1 8)

— = CmP^r (18)

where L is the tube length, r the radius of the tube and p is the mass per unit volume of the surrounding fluid, Cm is called the inertia coefficient which is a function of the geometry, and is discussed by Lamb (Lamb, 1932, 1945). If the moving tube is not infinitely long, the flow is three-dimensional and leads to smaller values of Cm (Moretti & Lowry, 1976). For a vibrating tube in a fluid region bounded by a circular cylinder, Stokes (Endres & Moller, 2009) has determined hydrodynamic mass per unit length as given by:

where Cm = — — , R is the outer radius of annulus and, ma = ржт2 , where p is fluid

m R2 _ r2

density, and r is the tube radius.

(Wambsganss, et al., 1974) have published a study on the effect of viscosity on Cm. Hydrodynamic mass M’ for a tube submerged in water was determined by measuring its natural frequency, fa, in air and, fw, in water. Neglecting the density of air compared to water, the following equation may be obtained from beam equation (Moretti & Lowry, 1976).

(20)

where p is the tube mass per unit length. The inertia coefficient, Cm, can be obtained from Equation 18. Figure 13 gives the results showing the variation of Cm with pitch-to-diameter ratios (Wambsganss, et al., 1974).

C„

P/D ratio

3.2 Damping

System damping has a strong influence on the amplitude of vibration. Damping depends upon the mechanical properties of the tube material, the geometry of intermediate supports and the physical properties of the shell-side fluid. Tight tube-to-baffle clearances and thick baffles increase damping, as does very viscous shell-side fluid (Chenoweth, 1993). (Coit et al., 1974) measured log decrements for copper-nickel finned tubes of 0.032 in still air. The range of most of the values probably lies between 0.01 and 0.17 for tubes in heat exchangers (Chenoweth, 1993). From (Wambsganss, et al., 1974), damping can be readily obtained from the transfer function or frequency response curve as

with / = /n (1) — /n (2),

where / is the resonant frequency and /N (1 and/N( 1 are the frequencies at which the response is a factor —^ of resonant response.

(Lowery & Moretti, 1975), have concluded that damping is almost entirely a function of the supports. More complex support conditions (non-ideal end supports or intermediate supports with a slight amount of clearance) lead to values around 0.04. From analytical point of view, (Jones, 1970) has remarked that in most cases, the addition of damping to the beam equation re-couples its modes. Only a beam, which has, as its damping function, a restricted class of functions can be uncoupled. (Chen et al., 1994) have found the fluid damping coefficients from measured motion-dependent fluid forces. (Pettigrew et al., 1986, 1991) outlines the energy dissipation mechanisms that contribute to tube damping as given in Table 6:

Type of damping	Sources
Structural	Internal to tube material
Viscous	Between fluid forces and forces transferred to fluid
Flow-dependent	Varies with flow regime.
Squeeze film	Between tube and fluid as tube approaches support
Friction	Coulomb damping at support
Tube support	Internal to support material
Two-phase	Due to liquid gas mixture
Thermal damping	Due to thermal load

Table 6. Energy dissipation mechanisms (Pettigrew et al., 1986, 1991) 4.3 Parameters influencing damping

(Pettigrew et al., 1986) further outlines the parameters that influence damping as given below:

The Type of tube motion. There are two principal types of tube motion at the supports, rocking motion and lateral motion. Damping due to rocking is likely to be less. Rocking motion is pre-dominant in lower modes. Dynamic interaction between tube and supports may be categorized in three main types, namely: sliding, impacting, and scuffing, which is impacting at an angle followed by sliding:

Effect of number of supports. The trend available in damping data referenced in (Pettigrew et al., 1986), when normalized give

£n = £N / (N -1) (22)

where £n is the normalized damping ratio, N is the number of spans, and £ is the damping ratio.

Effect of tube Frequency. Frequency does not appear to be significant parameter (Pettigrew et al., 1986).

Effect of vibration amplitude. There is no conclusive trend of damping as a function of amplitude. Very often, amplitude is not given is damping measurements (Pettigrew et al., 1986).

Effect of diameter or mass. Large and massive tubes should experience large friction forces and the energy dissipated should be large. However, the potential energy in the tube would also be proportionally large in more massive tubes. Thus, the damping ratio, which is related to the ratio of energy dissipated per cycle to the potential energy in the tube should be independent of tube size or mass (Pettigrew et al., 1986).

Effect of side loads. In real exchangers, side loads are possible due to misalignment of tube — supports or due to fluid drag forces. Side loads may increase or reduce damping. Small side loads may prevent impacting, and thus reduce damping, whereas large side loads may increase damping by increasing friction (Pettigrew et al., 1986).

Effect of higher modes. Damping appear to decrease with mode order, for mode order higher than the number of spans, since these higher modes involve relatively less interaction between tube and tube-support (Pettigrew et al., 1986).

Effect of tube support thickness. Referenced data in (Pettigrew et al., 1986) clearly indicates that support thickness is a dominant parameter. Damping is roughly proportional to support thickness. (Pettigrew et al., 1986) corrected the damping data line for support width less than 12.7mm such that

£nc =£n [ ^ J (23)

where L is support thickness in mm and £nc is the corrected normalized damping ratio.

Effect of clearance. For the normal range of tube-to-support diametral clearances (0.40mm — 0.80mm), there is no conclusive trend in the damping data reviewed (Pettigrew et al., 1986).

Design Recommendations (Pettigrew et al., 1986, 1991, Taylor et al., 1998)

Friction damping ratio in a multi-span tube (percentage)

where N is the number of tube spans, L is the support thickness, lm is the characteristic span length usually taken as average of three longest spans.

Viscous damping ratio

1 + (D / De )3 (1 — (D / De )2)2

Rogers simplified version of Chens’ cylinder viscous damping ratio (percentage) of a tube in liquid.

where p is the fluid density, m is the mass per unit length of tube (interior fluid and hydrodynamic mass), De is the equivalent diameter to model confinement due to surrounding tubes, D is the tube diameter, / is the frequency of tube vibration and и is

Squeeze film damping ratio

(For multi-span tube) £

Support damping

(Pettigrew, et al., 1991) has developed a semi-empirical expression to formulate support damping, using Mulcahys’ theory (Mulcahy, 1980).

(28) (TEMA, 6th Edition, 1978, TEMA, 7th Edition, 1988)

According to TEMA standards, £ is equal to greater of £1, and £2 (For shell-side liquids)

C = l^L

Wafn

where v is the shell fluid velocity, do is the outside tube diameter, po is the density of shell — side fluid, /n is the fundamental frequency of tube span, and Wo is the effective tube weight.

(For shell-side vapors)

where N is the number of spans, tb is the baffle or support plate thickness, and l is the tube unsupported span. A review of two-phase flow damping is presented by (Khushnood et al., 2003).