1.3.4.1 Two-Phase Boiling ModeV

If heat is input uniformly into a liquid coolant channel (Fig. 1.9), the temperatures of the fuel element surface Tcd and the coolant Tc increase as the coolant flows along the channel and the sequence of heating regions is as follows:

(a) Convective heat transfer region, in which the heat transferred q is given by

q — h(Tcd — Tc) (1.46)

Tc and Год increase linearly and Ted reaches a value just above the coolant saturation temperature.

(b) Highly subcooled boiling region, in which heat transfer is mainly convective. Bubbles form and collapse on the element surface and Tc increases.

(c) Slightly subcooled boiling region, in which heat transfer is partly by convection and partly by a boiling mechanism. Bubbles persist and are swept into the stream only to diminish in size as they move along the channel. The value of Tc increases until the saturation value is reached.

(d) Bulk-boiling region, in which the heat transfer q is given by

q = rj(Tcd — TB&ty (1.47)

Steam bubbles grow in the stream and Tc remains constant as all the heat goes to convert liquid into vapor.

(e) Superheated region after the liquid is fully converted to vapor and the vapor temperature Tc and the surface temperature Tcd now both rise again.

A pressurized water reactor (PWR). channel terminates in the slightly sub­cooled region whereas a boiling water reactor (BWR) channel extends into the bulk-boiling region. In a steam generator, a section will also include the superheated region.

+ See Tong (8b).

The mathematical representation of this two-phase situation uses the three conservation equations of energy, mass, and momentum to define the void fraction, the vapor velocity, and the pressure.

Energy

-fa ((?гП — fvihi + Qvfovhv) + — fa ((?i[l — fhi + QvKf) — b<f> + ± Gx

(1.48)

Mass

((?;[! —fvi + Qvfiv) + ~fa ((?iD ~f] + Qvf) = 0 (1-49)

Momentum

feD — />г2 + Qvfvv2) + feD — f]»i + 6vfav)

= (1’50>

However these equations presuppose some knowledge of the ratio of the vapor velocity to the liquid velocity vvjvt, the slip ratio, for which experi­mental correlations exist. The equations also assume that the two phases are in thermal equilibrium with each other. This is usually a good assump­tion except in very severe transients when the vapor and liquid might have to be considered separately. Note that the densities (, ov) and enthalpies (hi, hv) are all pressure dependent and the calculation is now very complex.

This model applies well to water and it may be used for water-cooled thermal reactors or in steam generator modeling. However sodium vapor­izes at high superheats and the two-phase situation is over very rapidly, resulting in almost immediate bulk boiling (see Section 1.3.4.3).