Two-phase natural circulation is the normal mode of coolant circulation in many advanced designs of BWRs. Typical examples are the simplified boiling water reactor (SBWR) and the advanced heavy water reactor (AHWR). Two-phase natural circulation is also the predicted mode of coolant circulation in most of the current designs of water cooled reactors during small break LOCA. Hence two configurations of loops as shown in Fig 2, one with a heat exchanger and another with a steam-water separator, are of interest to nuclear industry. For the development of the scaling relationships, only uniform diameter loops with adiabatic pipes operating without any inlet subcooling are considered here. The scaling laws for two — phase natural circulation can be obtained from the conservation equations of mass, momentum and energy applicable for homogeneous equilibrium flow.

dp 1 dW

dt A ds

d(ph) 1 d(Wh) _ 4q dt A ds D

H H

d(ph) 1 d(Wh) _ 0 dt A ds

d(rh) + ^ 0Wh) = _ 4U(Tsat — Ts)

dt A ds D

Ws, hs

Steam

FIG. 2a (left) Uniform diameter two phase loop with condenser. FIG. 2b (right) Uniform diameter two phase long with separator.

For the loop with the steam separator, instead of Eq. (23c) a point model of mass and energy equations is used, which assumes complete separation and thermal mixing in the separator. The feed water flow rate is assumed to be controlled to match the steam production rate. Integrating the momentum equation over the loop we get,

These equations are nondimensionalised using the following substitutions

Where tr=VtpL/Wss and Apss (always taken as positive) and Ahss are respectively the steady state density and enthalpy differences across the heated section. The nondimensional equations obtained are

with (fLO)2=1 for gingle-phage region and li = Li/Lt. In writing the above equation, the local preggure loggeg were congidered to be negligible.

4.1. Steady state flow

The gteady gtate governing equationg (all temporal derivativeg = 0 and wgg = 1) are da

dS

The LHS of the momentum equation becomeg zero for the uniform diameter loop ghown in Fig. 2a. For non-uniform diameter loopg with a tall riger (H in Fig. 2b) itg value becomeg negligible compared to the friction preggure drop. Hence,

% jP’dz = JNl RegA 2Rebg

For a loop with horizontal heater and cooler, it can be ghown that the <j" p * dZ = 1. Hence,

Where C=(2/p)r and r=1/(2-b). For laminar flow (b=1, p=64), C=0.03125 and r=1 and for turbulent flow (b=0.25 and p=0.316), C=2.87 and r=0.5714. The game relationghip can be obtained for the loop with vertical heater and cooler if we uge the elevation difference between the thermal centreg of the cooler and heater in place of the loop height in Grm. In

case, the value of the LHS in equation (33) is significant, then an explicit equation for Ress cannot be obtained. Instead, a polynomial in Ress can be obtained as

Where xex is the heater exit quality. Since xex=Q/Wsshfg without inlet subcooling, Eq. (36) can be rewritten as

For laminar flow, b=1, an explicit expression for Ress can be obtained. For turbulent flow, Ress can be numerically calculated from the above polynomial.

It can be shown that the Apss used in the Grm can be estimated as aexpfg+ (pL — pin). For no inlet subcooling, the density at the inlet is pL, then Apss=aexpfg where aex is the void fraction at the exit of the heater. If we use the homogeneous model for the evaluation of the heater exit void fraction, then aex=pL/[(pL-pG)+WsshfgpG/Q]. For the evaluation of the (fLO) , several homogeneous models are reported in the literature. If we use Owens (1961) model, then (fLO) =pL/ptp=1/[1-aex(1-pG/pL)]. Thus knowing the heater exit void fraction both Grm and NG can be evaluated. Although the homogeneous flow assumption was used in the derivation of Eq. (35), the difference in the velocities of the two-phases can be accounted by selecting appropriate models for aex and (fLO) .