The regions of instability and natural circulation are defined by the roots of Equations (3) and (8), using typical values for the loss coefficients. The parametric dependencies come by plotting maps in the convenient non-dimensional space of Np and Ns. Equation (10) provides the lower limit for inlet subcooling such that the flow will be stable for subcooling below this limit. Flows with the subcooling number greater than the critical subcooling number are potentially unstable, and the region of instability can then be characterized from the roots of Equation (8).

As is well known, the inlet loss coefficient has a stabilizing effect while the exit loss coefficient is destabilizing. As the inlet loss coefficient is increased, the unstable region moves to a higher subcooling number and is smaller; whereas as the exit loss coefficient is increased, the unstable region moves to a lower subcooling number and becomes larger.

The influence of the number of channels, n, is, where the loss coefficients for the parallel channels and components are reduced by a factor of (k/n), but the channels are often connected to a common separator, chimney and/or riser at the exit with ke = 3. Therefore, the unstable region is increased, and extends to lower values of the subcooling and phase change
numbers. Thus, in this example, the parallel channels are seen to be more unstable than a single channel, with the exit loss being more significant.

The parallel channel phenomenon is even more complex if the channels are not identical and have different cross-section areas, hydraulic diameters, flow resistance, power and flow (Popov et al, 2000). With channels with different geometry, losses and flow conditions operating in parallel, then the flow can have out-of-phase oscillations and flow diversion, and each channel has its own unstable boundary. If the power is gradually increased in a parallel channel system, keeping the inlet flow and subcooling fixed, the channels first go into subcooled and then saturated nucleate boiling. As power is further increased, the flow in the channels may oscillate as a result of a churn and slug flow regime (especially at low mass flow rates), and flow diversion from one channel to the other may occur.

A similar behaviour is expected when the flow is gradually reduced and the power maintained constant. The flow diversion from one channel into another may lead to local dryout. Therefore, as a first approximation, the metastable minimum and maximum point in the characteristic curve indicate the onset of flow diversion and dryout, respectively. It has indeed been observed that CHF could occur in the negative slope region between the minimum and maximum of the pressure drop vs. flow characteristic curve.

The results indicate that as the Froude number decreases, the flow becomes unstable, where zero Froude number is for horizontal flows, and negative Froude number is for downflow. On the other hand, the critical subcooling number decreases with the increase in the exit loss coefficient, implying that the stable region is smaller. Both these results have been also confirmed for density-wave instabilities.

Stability maps for vertical up, horizontal and down flows are shown in Np vs Ns space, in Figure 2, where the line for zero exit quality demarcates the region such that flows to the left are generally stable. The central ellipsoidal unstable region is bounded by curves that are the loci of two extrema of the pressure drop flow-rate curve. The first characteristic boundary of the unstable region close to the zero-quality line is the precursor of static instability, and represents the transition from the single-phase region to the two-phase region. The second characteristic boundary represents the transition from two-phase region to very-high-void region. One interesting result is that the unstable regions are asymptotic to a value of Np/Ns ~3, which is consistent with the asymptotic maximum power limits given by Equation (11). The line near Np/Ns ~1 also marks the onset of geysering instability in a closed vertical channel.