9.50. As shown in Fig. 9.6, fuel rod designs normally provide for a small annular gap between the uranium dioxide fuel pellet and the cladding. The gap is initially filled with helium, but during irradiation there is some dilution of the helium with fission-product gases. More important, how­ever, is the swelling and cracking of the pellet (§7.172) which tends to close the gap in a nonuniform manner. The interface heat-transfer problem therefore progresses from one involving conduction through the gas to a combination of conduction through the gas and across the interface be­tween two surfaces in partial contact. Also, the gap conductance changes with the power level and material effects related to fuel exposure (Chapter 7). Although analytical models have been developed based on the thermal conductance of a gas film of the thickness indicated by the roughness of the two surfaces, the results tend to be unreliable. Semiempirical computer models based on test data are therefore used for design purposes [2]. The gap conductance is also important in the calculation models required for safety evaluation (§12.128). For orientation purposes, the order of mag­nitude of the heat-transfer coefficient at the interface of irradiated oxide pellets and metallic cladding may be taken to be 6000 W/m2 • K.

Example 9.3. A PWR fuel rod (see Example 9.2), with pellets 8.19 mm in diameter, is clad with zircaloy 0.57 mm thick; the outer diameter of the clad rod is 9.5 mm. If the bulk (mixed-mean) coolant temperature is 315°C and the volumetric heat generation rate (power density) in the fuel is 3.20 x 108 W/m3, determine the temperature at the center of the fuel and at the outer surface of the cladding. The heat-transfer coefficient

CLADDING

at the cladding-coolant interface may be taken to be 34 kW/m2 • K. The thermal conductivity of the zircaloy is 13 W/m • K.

The temperature drop between the coolant and the cladding is obtained from the second term on the right of equation (9.22); thus,

Qa2

h tm ~ 2hb9

where

Q = 3.20 x 108 W/m3 a = 1/2 (8.19 mm) = 4.095 mm b = 1/2 (9.50 mm) = 4.75 mm

Hence,

_ _ (3.20 x 108)(0.004095)2 _

h tm ~ (2)(3.4 x 104)(0.00475) “

Since tm is 315°C, the cladding surface temperature is 315 + 17 ~ 332°C.

The temperature drop across the cladding is obtained from the first term on the right of equation (9.22); that is, where, as shown in Fig. 9.6, a’ is the inner radius of the cladding, i. e., 1/2(9.50) — 0.57 = 4.18 mm. Hence,

(3.20 x 108)(0.004095)2, 0.00475 h~k= (Ш) П 0^00418 = 26 C

(Use of the approximation for In (b/a’) in §9.36 footnote leads to essentially the same result.)

Next, consider the temperature drop At across the gas gap between the fuel and the cladding. This is given by an expression equivalent to the second term on the right of equation (9.22), except that b is replaced by a and h by the gap conductance. As stated in §9.50, the latter may be taken to be 6000 W/m2 • K; hence,

= (3.20 x 108)(0.004095)

2 ha (2) (6000)

The temperature drop across the fuel itself has been determined in Example 6.2 as 480°C; hence, the central fuel temperature is given by

t0 = 480 + 109 + 26 + 332 — 950°C.

This is an average temperature; it is expected that in some fuel rods the central temperature will be significantly higher. The results of the foregoing calculations are plotted in Fig. 9.7 with the central temperature value rounded off.