operating conditions must be considered including the following: 1) as‐designed operating conditions, from zero to 100% flow, 2) operating conditions with fouling of the tubes or crudding of the tube supports, and 3) other possible operating conditions (e.g., after chemical cleaning, system testing, etc.).
Fig. 2-2 Thermalhydraulic Analysis: Axial Grid Layout for a Typical Steam Generator with Preheater (Distances in Metres).
2.2.3 Comprehensive 3‐D Approach
For complex components such as nuclear steam generators and power condensers, a comprehensive three‐dimensional thermalhydraulic analysis is required. In such analyses, the component is divided into a large number of control volumes. The equations of energy, momentum and continuity are solved for each control volume. This is done with numerical methods using a computer code such as the THIRST code for steam generators (Pietralik, 1995). The numerical grid outlining the control volumes for the analysis of a typical steam generator is shown in Fig. 2-2. The grid must be sufficiently fine to accurately predict the flow distribution along the tube. Some typical thermalhydraulic analysis results are shown in Fig. 2-3 for the U‐bend region of a steam generator. For flow‐induced vibration analyses, the results must be in the form of pitch flow velocity and fluid density distributions along a given tube. These distributions constitute the input to a flow‐induced vibration analysis of this particular tube. Figure 2-4 shows pitch flow velocity and fluid density distributions for an example condenser tube.
Fig. 2-3 Flow Velocity Vectors in the Central Plane of a Typical Steam Generator U‐Bend Region.
Fig. 2-4 Gap Cross‐Flow Distribution Along a Typical Condenser Tube.
Fig. 2-5 Flow Pattern Map for Two‐Phase Flow Across Cylinder Arrays Using Flow Pattern Boundaries (Grant and Chisholm (1979) and Axes Parameters from McNaught (1982)).
2.2.4 Two‐Phase Flow Regime
Some knowledge of flow regime is necessary to assess flow‐induced vibration in two‐phase flow. Flow regimes are governed by a number of parameters, such as surface tension, density of each phase, viscosity of each phase, geometry of flow path, mass flux, void fraction and gravity forces. Flow regime conditions are usually presented in terms of dimensionless parameters in the form of a flow regime map. Grant (1975) and Grant and Chisholm (1979) used available data to develop the flow regime boundaries shown in Fig. 2-5. The axes on the Grant flow regime map are defined in terms of a Martinelli parameter, X, and a dimensionless gas velocity, Ug. The Martinelli parameter is formulated as follows:
(2‐7)
where μℓ and μg are the dynamic viscosity of the liquid and gas phases, respectively. The dimensionless gas velocity is defined as follows:
(2‐8)
where
The Grant map of Fig. 2-5 shows three flow regimes: spray, bubbly and intermittent. The terms spray and bubbly are used loosely here. Perhaps they would be more appropriately defined as “continuous flow” covering the whole range from true bubbly flow to wall‐type flow to spray flow. Intermittent flow is characterized by periods of flooding (mostly liquid) followed by bursts of mostly gas flow. As discussed by Pettigrew et al (1989a) and Pettigrew and Taylor (1994), this is an undesirable flow regime from a vibration point‐of‐view. Thus, intermittent flow should be avoided in two‐phase heat exchange components and, particularly, in the U‐bend region of steam generators.
Flow regime is discussed in more detail in Chapter 3.
2.3 Dynamic Parameters
The relevant dynamic parameters for multi‐span heat exchanger tubes are mass and damping.
2.3.1 Hydrodynamic Mass
Hydrodynamic mass is the equivalent dynamic mass of external fluid vibrating with the tube. In liquid flow, the hydrodynamic mass per unit length of a tube confined within a tube bundle may be expressed by
Fig. 2-6 Hydrodynamic Mass in Two‐Phase Cross Flow: Comparison Between Theory and Experiments. (Note that mℓ is the hydrodynamic mass per unit length in liquid.)
where De is the equivalent diameter of the surrounding tubes and the ratio De/D is a measure of confinement. The effect of confinement is formulated by
(2‐10)
for a tube inside a triangular tube bundle (Rogers et al, 1984). Similarly, for a square tube bundle (Pettigrew et al, 1989a) confinement may be approximated by
(2‐11)
The hydrodynamic mass of tube bundles subjected to two‐phase cross flow may be calculated with Eq. (2-9)
