The other possibility is that the bubble in the channel would be relatively large and would either lodge somewhere or would pass through the core transiently insulating the fuel pin as it passes through.

Studies have shown that the passage of single bubbles at normal sodium velocities, assuming complete insulation of the pin surface under the bubble, results in perturbations of tens of degrees rather than hundreds. Thus no sudden failure is likely (5b). If the bubble were to lodge in a single position for a comparatively long time, then the surface temperature could increase, unless the axial conduction or circumferential conduction paths were ade­quate in removing the heat from the surface under the bubble.

Axial conduction is not efficient and thus most heat is removed circum­ferentially through the cladding from beneath the bubble. Whether this latter path is efficient or not depends on the circumferential coverage of the bubble. For small bubbles the heat is removed through the cladding faster than it is input from the fuel within the cladding, but for larger bubbles the. heat input is the faster process. Thus there is a critical coverage which may be obtained from a comparison of the time constants for heat input into the cladding and heat transport circumferentially through the cladding.

For a bubble which covers an angle в of the fuel pin surface and assuming no heat removal (see Fig. 4.3), the time constant for the input of heat into the cladding below the bubble is given by

Tin = fefCf&dCcd^2 fa)l(p{C{R + 2gcdccdt) K{(RlV 3 + b) (4.9)

Assuming no heat input, the time constant for heat removal along the cladd­ing in a circumferential direction is given by

Tout = (QcicciR*e*)/4Kci (4.10)

The critical angle of coverage вст is defined as that angle when these time constants are equal. Below this angle of coverage the cladding surface temperature rises only moderately above normal values, whereas above this value cladding temperatures rise considerably more. The critical angle 6CI is given by

всг = (4 e(CfKcd dt)1/2/[(fttc(R + 2pcdccdl) Kt(R/f3 + P)]1/2 (4.11)

An approximate expression may be derived as

0cr = 0.268(8A’cjt/A’fT?)172

For a fuel pin of 0.25-in. diameter, cladding thickness of 0.0125 in., and with oxide fuel clad in stainless steel, the critical angle is 94°.

The temperature rises beneath such a bubble may be calculated using one of the three dimensional heat transfer codes which are available, TOSS or TRUMP (see the Appendix). The models used in these codes are the basic heat transfer equations detailed in Section 1.2 but arranged on a mesh that connects in three dimensions, making computer solution a ne­cessity. Many of the difficulties involved in these calculations are connected with the spatial finite difference approximations used to represent conduc­tion from one point in the mesh to another.

If the complete pin were blanketed by a bubble and the angle of coverage were 360°, then axial conduction would become important. Studies have shown that, for the above fuel pin, approximately an inch would have to be insulated by the bubble before failure of the fuel pin cladding could occur. For bubbles less than 1 in. in extension only moderate cladding temperature rises would result.