GENERAL STATIC INSTABILITY THEORY AND THE ONSET OF CHF

We have derived two results, which enable both the stability and natural circulation characteristics to be evaluated analytically and with the minimum of computation.

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.

Подпись: <9AP ~dG Подпись: 0 Подпись: (7)

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),

This condition, when applied to Equation (1) and with the vapor to liquid density ratio being small, leads to the following characteristic Equation (8), which is derivable after many pages of straightforward but extensive algebra:

image016(8)

where Nf, Ns, Nfr, and Np are again the Froude, Subcooling, Friction and Phase Change numbers, respectively. In addition, the k-parameters are loss coefficients normalized with the Friction Number. (The last term in brackets is corrected for a typographical error in the earlier paper (Rohatgi and Duffey, 1994) which omitted the number 2.)

It is important to note that arising naturally from the analysis, the instability region is bounded by the two roots of Equation (8), where the pressure drop versus flow rate is a minimum, maximum or point of inflection. The region of instability for a parallel channel system as bounded by these roots, which correspond to low (subcooled boiling) and high (saturated boiling) vapor qualities. This result has been rederived and confirmed by (Babelli et al, 1995), who retained the vapor to liquid density ratio, and the convective acceleration term, to derive
a "corrected result" more complex than Equation (8). The overall effect of the corrections is generally small. The two roots of the quadratic have been termed the first and second analytic instability lines by (Babelli et al, 1995), corresponding to the low and high quality flow states respectively.

The conditions for existence of positive real solution in Equation (8) which is quadratic in Np and Ns, are as follows:

image017Подпись: and (9)

(10)

image019 Подпись: (11)

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, for small,

or

Nr_n N?_ — — Qor—— 2