There are a number of instabilities that may occur in two-phase systems. These may be classified into static and dynamic instabilities [Lahey andPodowski (1989)]. Examples *of static instabilities* include: *flow excursion* (i.e., Ledinegg) *instabilities, flow regime relaxation instabilities, geysering* or *chugging instabilities* and the *terrain-induced instabilities* that may occur in off-shore oil well lines. Similarly, *dynamic instabilities* include: *density-wave oscillations, pressure drop oscillations, flow regime-induced instabilities* and *acoustic instabilities*.

Of these instability modes, the most important, and most widely studied, have been *Ledinegg instabilities* [Ledinegg (1938)] and *density-wave oscillations* (DWOs). While the subsequent discussion will be focused on boiling systems, it should be noted that similar instabilities may also occur in condensing systems [Lahey and Podowski (1989)].

A typical *boiling loop* includes a heated channel (or channels), an unheated riser, a condenser, a downcomer (in which a pump may be installed) and a lower plenum. A dynamic force balance on the boiling loop yields:

where, Δp_{ext} is the impressed pressure rise in the system (e.g., due to a pump) and Δp_{sys} is the pressure drop of the system at flow rate w. The hydraulic inertia of the loop (I) is given by

Linearizing Equation (1), we obtain

which has a solution given by

We note that Equation (4) implies that a flow excursion will occur if

Figure 1 shows four cases for a typical low pressure boiling loop. Case 1 is for a positive displacement pump. In this case ∂(Δp_{ext})/∂w = –∞, thus Equation (5) is never satisfied and thus the loop is stable at operating point 1. Case 2 is the so-called parallel channel case in which a constant Δp_{ext}, is imposed across each boiling channel. For this case ∂(Δp_{ext})/∂w = 0, thus Equation (5) implies that operating point 1 is unstable (however operating points 2 and 3 are both stable). Similarly, cases 3 and 4 represent the situation in which a centrifugal pump is used in the loop. We note that case 3 is unstable while case 4 is stable.

While Ledinegg instability is known to be a problem in low pressure boiling systems, an increase of the system pressure, or an increase in the inlet orificing in the channel, can stabilize the system.

The quantification of density-wave oscillations requires an analysis of the mass, momentum and energy conservation equations of the boiling system.

A detailed description of the analytical procedure has been given previously by Lahey and Podowski (1989) and will not be repeated here. The essence of the analytical procedure is to determine first the neutral stability boundaries using a linear stability technique, then a nonlinear analysis is performed to identify bifurcation phenomena and to assess the dynamic response of the boiling system.

In a linear stability analysis, the mass and energy conservation equations may be integrated in the axial direction either analytically [Lahey and Moody (1993)] or using nodal techniques [Taleyarkhan et al. (1985)]. The results of this analysis are then combined with the integrated and linearized momentum equation to satisfy the pressure drop boundary condition impressed on the boiling channel(s) or loop. Laplace transforming the result, we obtain the following characteristic equation, for either the loop (k = L) or the heated channel (k = H),

System stability is determined by solving for the roots, s, of Equation (6). If we have any complex roots (s) having positive real parts the system is unstable. Also, we note that Re(s) = 0 defines the neutral stability boundaries.

For a boiling loop (k = L) we have

where, Г(s) and П(s) are the transfer functions of the single-phase and two-phase parts of the loop, respectively, and and are the Laplace transformed inlet velocity and the volumetric power perturbations.

In a nuclear reactor there are also feedback loops associated with temperature- and void-reactivity feedback. This latter transfer function may be denoted, T(s). As should be expected, the stability of a *nuclear-coupled* system is in general different from that given by Equation (6).

Figure 2 is a typical block diagram for a *boiling water nuclear reactor* (BWR). It can be shown [Lahey & Moody (1993)] that the characteristic equation associated with this block diagram is

where Ф(s) is the so-called zero-power point neutron kinetics transfer function [Weaver (1963)] and C_{α} is the BWR's void-reactivity coefficient. As before, the stability of a BWR can be determined by solving for the roots, s, of Equation (8). A typical linear stability map is given in Figure 3 in terms of the nondimensional phase change number (N_{pch}) and the subcooling number (N_{sub}).

Finally, it should be noted that time domain evaluations may be performed with the nonlinear conservation equations leading to Hopf bifurcations (e.g., limit cycles) and, in some cases, chaos [Garea et al. (1994)].

There are many other things that could be said about two-phase instabilities. Indeed there is a voluminous literature on this subject [Lahey and Drew (1980)]. Nonetheless, this introduction has given the essence of two-phase instability analysis and presented some of the key references so that the interested readers may develop a more indepth understanding.

#### REFERENCES

Garea, V. B., Chang, C. J., Bonetto, F. J., Drew, D. A. and Lahey, R. T., Jr., (1994) The analysis of nonlinear instabilities in boiling systems, Proceedings of the International Conference on New Trends in Nuclear System Thermohydraulics, Pisa, Italy, May 30-June 2.

Lahey, R. T., Jr. and Drew, D. A. (1980) An Assessmenl of the Literature Related to LWR Instability Modes, NUREG/CR-1414.

Lahey, R. T., Jr. and Moody, F. J. (1993) The Thermal-Hydraulics of a Boiling Water Nuclear Reactor, ANS Monograph.

Lahey, R. T., Jr. and Podowski, M. Z. (1989) On the analysis of various instabilities in two-phase flows, *Multiphase Science and Technology*; Vol. 4, Hemisphere Publishing Corp.

Ledinegg, M. (1938) "Instability of flow during natural and forced circulation," *Die Wärme*, 61, 8.

Taleyarkhan, R., Lahey, R. T., Jr. and Podowski, M. Z. (1985) An instability analysis of ventilated channels, *J. Heat Transfer*, 107.

Weaver, L. E. (1963) A System Analysis of Nuclear Reactor Dynamics, ANS Monograph.

#### References

- Garea, V. B., Chang, C. J., Bonetto, F. J., Drew, D. A. and Lahey, R. T., Jr., (1994) The analysis of nonlinear instabilities in boiling systems, Proceedings of the International Conference on New Trends in Nuclear System Thermohydraulics, Pisa, Italy, May 30-June 2.
- Lahey, R. T., Jr. and Drew, D. A. (1980) An Assessmenl of the Literature Related to LWR Instability Modes, NUREG/CR-1414.
- Lahey, R. T., Jr. and Moody, F. J. (1993) The Thermal-Hydraulics of a Boiling Water Nuclear Reactor, ANS Monograph.
- Lahey, R. T., Jr. and Podowski, M. Z. (1989) On the analysis of various instabilities in two-phase flows,
*Multiphase Science and Technology*; Vol. 4, Hemisphere Publishing Corp. - Ledinegg, M. (1938) "Instability of flow during natural and forced circulation,"
*Die WÃ¤rme*, 61, 8. - Taleyarkhan, R., Lahey, R. T., Jr. and Podowski, M. Z. (1985) An instability analysis of ventilated channels,
*J. Heat Transfer*, 107. - Weaver, L. E. (1963) A System Analysis of Nuclear Reactor Dynamics, ANS Monograph.