Bubbles within coolant channels may affect the heat transfer in two ways. If the bubbles are small and dispersed and spread evenly throughout the coolant, they have a homogeneous effect of macroscopically decreasing the density of the coolant. If the bubbles are larger, they may be considered individually as insulating a portion of the fuel pin either in a stationary manner or in a transient manner as the bubble moves up the core.

4.2.2.1 Homogeneous Effects+

If the bubbles are small and dispersed, the density is effectively reduced. The Nusselt number is given by an expression of the form

Nu = 4.8 + 0.025 P°-8 (4.3)

Thus as Nu = hDjk and Pe = QDv(cP/k) this equation directly leads to an evaluation of the heat transfer coefficient h, which is much simplified if one assumes that the density of gas is negligible compared to the density of sodium.

With gas in the sodium coolant, the effective density is given by Eq. (4.4) in which a is the void fraction. The average velocity of the coolant and the

+ See Hori and Hosier (52).

heat capacity cP’ are given by Eqs. (4.5) and (4.5a).

For a two-phase mixture the thermal conductivity for small void fractions is given (5a) by Eq. (4.6). This equation is valid up to a void fraction of 0.5

k! = £(1 — a)/(l + £a) (4.6)

Thus substituting these values for k’, v’, cv’, and q’ for the single phase values in Eq. (4.3) the homogeneous coolant heat transfer equation becomes

h’ = A[(l — a)/(l + £a)][4.8 + 0.025 P™( 1 + Jo)0-8] (4.7)

In the limit of the validity of this equation, for a void fraction of 50%, the heat transfer is reduced to about a third of its original value.

The fuel temperature is related to the coolant temperature in steady — state conditions through the heat-transfer coefficient as

fuel pin surface temp. = coolant temp. + power/А’ (4.8)

Thus the temperature difference between the surface of the fuel pin and the coolant will increase by something less than three times. Thus, even with a void fraction of 50% the surface temperature on the hot pin cladding might increase from 1200 to 1500°F and failure is unlikely. In practice, this volume of bubbles would not be possible.