Flow instability in the boiling channels of fossil or nuclear plants would soon lead to boiler tube or fuel pin ruptures from the thermally induced stresses. Experiments show that channel flow oscillations with a period of 1 to 10 s can exist under constant inlet and outlet pressures. A resolution of this paradox is obtained by considering the transport delays and changes of thermodynamic phase along a boiling channel [80,117]. Qualitatively, an inlet mass flow perturbation dW 1sin vt in Figure 3.4 persists over the largely incompressible liquid-phase region to produce a simultaneous differential pressure drop of dPL sin vt. However, due to its compressibility, the average acceleration of the two-phase region and its corresponding pressure perturbation dP2f suffer a significant delay. A similar argument applies a fortiori to the steam region, whose average acceleration and differential pressure change dPs are still further delayed. The phasor diagram in Figure 3.4 reveals qualitatively that a flow oscillation can exist with no change in
the differential pressure across a boiling channel. Flow stabilization can then only be achieved by inserting inlet ferrules (gags), which increase the liquid-phase component of a differential pressure change. As can be inferred from Figure 3.4, this artifice pulls the Nyquist diagram away from the critical (—1,0) point. Though feedpump power is relatively small,[36] quite small changes (e. g., 0.1%) in station efficiency are material [80], so the ferrules must be designed to be sufficient for purpose and little more.[37] Engineering experience and comprehensive non-linear simulations are now shown to simplify a quantitative analysis of the problem.

In response to an inlet-flow perturbation vector d W 1 (t) in Figure 3.5 define

Primary inlet mass flow perturbation Primary inlet temperature perturbation Water-side inlet mass flow perturbation Water-side inlet temperature perturbation Water-side inlet pressure perturbation

Upper plenum

SP,.,„ = о

Figure 3.5 Parallel Channel Stability Model

and the incremental outlet flow vector of the boiling channel by

Primary outlet mass flow perturbation Primary outlet temperature perturbation Water-side outlet mass flow perturbation Water-side outlet temperature perturbation Water-side outlet pressure perturbation

which are related by a matrix transfer function T(s). During the observed period of flow oscillations, water-inlet temperature is effec­tively buffered by a large mass stored in the feed train [141], and inlet pressure by the feedpump circuit. Also at high subcritical pressures, heat transfer and thermodynamic states along a channel are essentially unaffected [64] by the relatively small pressure changes, and those across a ferrule simply add to the overall differential pressure pertur­bation [117]. Both inlet and outlet primary-side variables are main­tained effectively constant by virtue of their: thermal capacities, pump speed, and reactor reactivity settings.5 Accordingly the perturbed inlet mass flow vector to a channel reduces to

u = (Water-side inlet mass flow perturbation) (3.3)

Thus the outlet water-side pressure perturbation dP0 is given for practical purposes by the SISO transfer function relationship

dP0 = T53 (s)dW 1 (3.4)

5 This conclusion applies to both fossil and nuclear plants.

With large amplitude mass flows, the pressure drop across a ferrule is [304]

Pf = Pf (W2/pL; Ferrule Geometry) (3.5)

but because the density of liquid-phase water is essentially constant under all normal operating conditions, incremental pressure and mass flow changes for a ferrule are related by

dPf = —KfSWi; Kf > 0 (3.6)

where Kf is a constant specific to a ferrule’s geometry and the inlet flow Wi at the selected output power. It follows from Figure 3.5 that no change in overall differential pressure occurs when

0 = SPout = — Kf SW i + T53 (s)SW i + SPnoise (3.7)

where the noise term SPnoise arises from fluid turbulence and variations in pump speed induced by Grid-frequency fluctuations. Incremental flow stability by a choice of ferrule geometry can therefore be engi­neered from

By analogy with a SISO unity feedback system, Davis and Potter [83] originally in the context of SGHWR described —T53(s) as the “open loop” transfer function, and by analytical linearization derived transfer functions for the three different water-phase zones in Fig­ure 3.4. Suitable ferrule geometries to cope with different load factors and flow distribution in the lower plenum were then selected by Nyquist diagram techniques. As an alternative, Knowles [117,128] perturbs comprehensive non-linear simulations to achieve the same end: but on the basis of the above more rigorous state vector formulation. Nevertheless, the original simplified analysis is sound and it illustrates the considerable simplification achievable by engi­neering insight and industry specific experience. However, as with all linearized models, confirmation by a full non-linear simulation is absolutely necessary.