In designing heat removal systems, we often rely on both single and two-phase natural circulation. The fluid heats and expands, and may boil and the two-phase flow generates an increased pressure drop. The inlet flow is derived from a pump, or a downcomer with a hydrostatic head, so that when there are many channels in parallel, as in a reactor, heat exchanger or condenser, the boundary condition is a constant pressure drop.

We can derive a formulation for the natural circulation flow, which includes the subcooled length and the effect of flow on the loss coefficients; is consistent with homogeneous stability analysis, and extends previous work. This result can then be coupled with the instability treatment to determine the natural-circulation limit on instability.

Consider any natural-circulation system as a loop or U-tube, where the driving force for the flow is due to the gravitational head difference between the water in the downcomer and that in the boiling channel and riser. In the steady state, which is sufficient for our discussion, this head difference is balanced by the pressure losses incurred by the flow and venting of the boiling two-phase flow, so in a natural circulation loop, the pressure drop boundary condition is zero around the loop. Thus, the net gravitational hydrostatic head is exactly balanced by the friction and form losses in the single and two-phase flow. Consider a uniformly heated channel of equivalent hydraulic diameter, D, with a subcooled inlet flow rate, G. To a good approximation, we take the flow as steady and incompressible. The momentum equation in the vertically inclined channel is expressed in the usual way in Equation (1).

og з(g2/ p) 3t 3z	f"(f+k )G 2/p )+ )p cosq
(1)
By integrating the momentum (Equation (1)) around the loop, we have the sum of the pressure drop components (Rohatgi and Duffey, 1994),
L-Zx tf [A; k, 2ZH Pt Д* J 21 Pt aJ s{zjPt + (g — zj)ps — ZxPt — t fe — iz — 4W} =0 Here we have where, by definition, the subcooled length, is for Xin< 0,
(2)
hfpXinAGL
(3)
Q
and, G, pme, ki, ke, Xin, Xe and Xa are mass flux, mean mixture density in the boiling section, mixture density at the exit, inlet and exit loss coefficients, inlet and exit qualities, and mean quality of the boiling region, respectively. We have defined mean channel and exit densities given by,
	Vg + Xa Vg Vp + Xe vp Vg + Xa Vg
Equation (2) is now cast in phase-change and subcooling number form (Rohatgi and Duffey, 1994). The resulting expression is, after much algebra and neglecting the small terms with the vapor to liquid density ratio:
where,
(4)
(4.1)
Friction Number Nfr
(4.2)
JL_ 2D
(4.3)

Phase Change (Zuber) Number N? — ———

AGhfgPg

and

Downcomer Number

(4.5)

Equation (4) is quadratic in Np and Ns and the roots provide the possible set of solutions for all liquid single and boiling two-phase flow. We can easily obtain the power required for given flow rate and core inlet subcooling from Equation (4), and vice versa. The Froude number will vary with the flow direction; the downcomer dimension, L*, is a design parameter; with the non-dimensional length being the ratio of the channel to downcomer heights L*, and typical values can be taken for the loss coefficients.

The natural circulation map describes the two allowed power and flow relationships for a given heated channel length to downcomer head ratio, L*, for given loss coefficients and Froude number. The two solutions correspond to single (liquid only) and two-phase flow for the same downcomer number at a given subcooling. The solutions of interest in a boiling loop are in the two-phase region. The intersection at zero subcooling number, for a large exit loss, is given very nearly by,

Np (Ns=0) = (2Nf /keNfrL*) -1 (5)

It is clear that the dependency between Np and Ns is nearly linear for a given set of loss coefficients and downcomer height, L*. Thus, the power to flow ratio is uniquely defined, and the curves describe the allowed flow states for a particular set of design values for the loss coefficients and relative downcomer height, L*. It is evident the flow can be bistable about the saturated line.

Since the flow is effectively bistable (double valued) for a given set of conditions, theoretically one can have single or two phase flow for the same pressure drop, an effect that is often observed during flow reversals and instabilities too. The effect of the relative driving head is shown by the following figure, where for a given set of loss coefficients the solutions to Equation (4) is shown for variations in L*, the relative heated length to downcomer height.

Now variations in downcomer height also can correspond to variations in total loop fluid inventory. The effect of changing total loop inventory on flowrate is determined by the shifts in the void in the boiling regions. So initially decreasing the inventory leads to an increasing flowrate, as the driving head increases since there is more boiling on the “hot” side (the riser or heated section). Eventually as the void increases, the flow reaches a maximum and then decreases as the driving head falls in the “cold leg” or downcomer.

Physically, the argument can be shown in more detail, and to illustrate the effects, Equation (2) can be solved assuming steady flow, taking a separator elevation equal to the downcomer height, and mean densities taken for the two-phase regions in the hot (riser, R) and cold (downcomer, D) regions. Lumping the loss terms together into an “effective” loss coefficient (Duffey and Sursock, 1987) it is possible to derive simple equations that describe the general behavior of the flow (and agree with the trends in the data). Thus, the loop flowrate is approximately given by:

where,

is the maximum possible flow if it were all liquid, and Ф is a two-phase friction multiplier, and I is the fractional liquid inventory in the loop, W1 the single-phase flowrate, VR is a measure of the hot side fractional volume to the total system volume, and Y is a simple interpolation function for a smooth flow transition. Since it is evident that the two-phase mass flow goes as (1-I) , we could maximize the flow in a design by adjusting the values of the losses, relative volumes and heated lengths.

It is well known that a two-phase natural circulation flow can exhibit instabilities over certain regimes (Gulshani et al, 1995) so we now examine that phenomenon and give some relevant analytical results.