Static Instabilities

In Chapter 3 the steady-state hydrodynamic aspects of two-phase flow were discussed and reference was made to their potential for instabilities. The instability of a system may be either static or dynamic. A flow is subject to a static instability if, when the flow conditions change by a small step from the original steady-state ones, another steady state is not possible in the vicinity of the original state. The cause of the phenomenon lies in the steady-state laws hence, the threshold of the instability can be predicted only by using steady-state laws. A static instability can lead either to a different steady-state condition or to a periodic behavior (Boure et al., 1973). A flow is subject to a dynamic instability when the inertia and other feedback effects have an essential part in the process. The system behaves like a servomechanism, and knowledge of the steady-state laws is not sufficient even for the threshold prediction. The steady-state may be a solution of the equations of the system, but is not the only solution. The above-mentioned fluctuations in a steady flow may be sufficient to start the instability. Three conditions are required for a system to possess a potential for oscillating instabilities ... 

Simple static instability. Flow excursion (Ledinegg instability) involves a sudden change in the flow rate to a lower value. It occurs when the slope of the channel demand pressure drop-versus-flow rate curve (internal characteristic of the channel) becomes algebraically smaller than the loop supply pressure drop-versus-flow rate curve (external characteristic of the channel). The criterion for this first-order instability is... 

Compound dynamic instabilities as secondary phenomena. Pressure-drop oscillations are triggered by a static instability phenomenon. They occur in systems that have a compressible volume upsteam of, or within, the heated section. Maul-betsch and Griffith (1965, 1967), in their study of instabilities in subcooled boiling water, found that the instability was associated with operation on the negative-sloping portion of the pressure drop-versus-flow curve. Pressure drop oscillations were predicted by an analysis (discussed in the next section), but because of the... 

As indicated previously, static instabilities, being induced mostly by primary phenomena, can be predicted by using steady-state criteria or correlations. Therefore, the threshold of static instability can be predicted by using steady-state evaluations. 

Check the static instabilities by steady-state correlations, to avoid or alleviate the primary phenomenon of a potential static instability, namely, boiling crisis, vapor burst, flow pattern transition, and the physical conditions that extend the static instability into repetitive oscillations. 

Bi, H. T. and Zhu, J. (1993). Static Instability Analysis of Circulating Fluidized Beds and Concept of High-Density Risers. AlChE J., 39,1272. 

Figure 4-15, however indicates that an exclusion of static instability is not automatically sufficient for the safety assessment of the process under normal operating conditions. In this example operating points with high sensitivity values still exist. They are in the range of medium conversion values. Figure 4-17 shows the corre-spondii sensitivity values in addition to the previous information. 

This need not be considered for the modified process as the extended sensitivity analysis already excludes any tangential point with the dynamic stability limit, this way completely excluding any static instability. 

Static instabilities induce a shift of the equilibrium point to a new steady-state point Ledinegg instability, boiling crisis, bumping, geysering, or chugging are all static instabilities since they can be analyzed using only stationary models. 

In the atmosphere, the same mechanism leads to static instability. However, the density of air changes substantially (about an order of magnitude in one vertical scale... 

In this moist convective instability, the same derivation for static instability applies except that the 0 profile is the relevant thermodynamic parameter, rather than the 0 profile. Thus the condition for moist convective instability... 

A static instability occurs when a small perturbation from original steady-state flow leads to a new stable operating condition which is not in the vicinity of the original state. The mechanism and the threshold conditions are predicted using steady-state characteristics of the system. Pressure drop characteristics of a flow channel, nucleation properties, and... 

Incidentally, a theory having orientation as an Internal variable will permit a static instability even where the stress-strain curve is increasing. For example, the energy function... 

Bi HT, Zhu JX. Static instability analysis of circulating fluidized beds and the concept of high-density risers. AIChE J 39 1272-1280, 1993. 

These "static" instabilities, which arise as a result of pressure difference across a piston or from an axial movement of the piston, can be compensated by rotating the piston. 

Once more was known about the consequences of a pressure difference and an axial (translatory) movement of a grooved piston, it was possible to look for a practical construction for a piston with a hydrodynamic bearing in a compressor. Obviously, the construction would have to prevent negative stiffness ("static" instability) from occurring. 

This investigation has resulted in a great number of constructive measures that can prevent "static" instability, see lit. [8]. Of these measures, one is described here. 

Chatoorgoon, V., 2013. Non-dimensional parameters for static instability in supercritical heated channels. International Journal of Heat and Mass Transfer 64, 145—154. 

Swapnalee, B.T., Vijayan, P.K., Sharma, M., PiUdiwal, D.S., 2012. Steady state flow and static instability of supercritical namral circulation loops. Nuclear Engineering and Design 245, 99-112. 

The divergence is a statical instability where the system response grows exponentially. Flutter, on the other hand, is a dynamical instability and involves system vibration with growing amplitude [48]. 

The requirement for both dynamic and static instabilities is that the increase in the two-phase pressure drop should be either equal to or greater than the decrease in the single-phase pressure drop as the inlet flow decreases. The relevant limit is actually the static (non-linear) instability boundary, which may lead to CHF, has been called the "zeroth mode" of dynamic instability. Thus, in dynamic dispersion-type analysis, it corresponds to the time-independent, zero-frequency (or infinite wave number), real wave number case which, corresponds precisely to the homogeneous equilibrium limit for the flow. In non-linear (called excursive instability ), the channels could switch from one flow rate to another while maintaining the same total pressure drop. When non-linearly unstable, the channel flow fluctuates, or reverses, and dryout can ensue. ... 

The static limit is the non-linear limit of conditionally instability, where departure from nucleate boiling or critical heat flux will occur at low and high qualities, respectively. There are sufficient data in the literature which show that instability in multiple channels precedes the limit of classic single channel (mass-flow controlled) dryout (Mathison, 1967) (D Arcy, 1967). This differs from the result for the zero frequency condition, which can only be written as a cubic in, (Ns / Np), and does not give a critical subcooling number. The condition of static instability in parallel channels is the Ledinegg condition (Saha et al, 1976) (Duffey and Hughes, 1991),... 

The above equation is also quadratic in Ns and provides a lower bound for the static instability. For large values of Np and Ns, the asymptotic form of Equation (8), has the limits,... 

---