*3.1.1 Fuel pellet conductivity and temperature calculation*

PHWR fuel is made of ceramic containing *UO*<sup>2</sup> with 0.7% U-235. It has poor heat conductivity properties as compared to carbide or metallic uranium. The heat transfer coefficient is dependent upon temperature as well as fission product accumulating inside the pellet. Moreover, as the fuel undergoes irradiation, cracks develop which changes the conductivity. A widely used correlation for the calculation of temperature-dependent pellet conductivity is as follows [11].

$$\mathbf{0} < \mathbf{T} \le \mathbf{1} \mathbf{650}^{\mathbf{o}} \mathbf{C}$$

$$\mathbf{h}\_{\mathbf{P}} = \eta \frac{\mathbf{B}\_{1}}{\mathbf{B}\_{2} + \mathbf{T}} + \mathbf{B}\_{3} \mathbf{e}^{(\mathbf{B}\_{4} \mathbf{T})} \tag{2}$$

$$\mathbf{1650} \le \mathbf{T} \le 294 \mathbf{0}^{\mathbf{o}} \mathbf{C}$$

$$\mathbf{h}\_{\mathbf{P}} = \eta \left[ \mathbf{B}\_{5} + \mathbf{B}\_{3} \mathbf{e}^{\mathbf{B}\_{4} \mathbf{T}} \right] \tag{3}$$

$$\mathbf{I} = \mathbf{I} \tag{4}$$

$$\eta = \left[ \frac{\mathbf{1} - \mathfrak{B} \left( \mathbf{1} - \frac{\rho}{\rho\_{\rm TD}} \right)}{\mathbf{1} - \mathfrak{B} (\mathbf{1} - \mathbf{0}.95)} \right] \tag{4}$$

Where <sup>η</sup> is the porosity factor and <sup>β</sup> <sup>¼</sup> <sup>2</sup>*:*<sup>58</sup> � <sup>0</sup>*:*<sup>58</sup> � <sup>10</sup>�<sup>3</sup> � T. The constants for a different fuel types are shown in **Table 1**.

This correlation is based on the data pooled from ten sources and an analytical expression is generated based on this data. The integral of UO2 thermal conductivity between 0°*C* and the melting point 2850°*C* is analytically determined in MATPRO. Assuming that the electronic contribution B3eð Þ B4T has the value of <sup>2</sup> � <sup>10</sup>�3w*=*cmKat 1500°C, a least-squares value of 97w*=*cm is obtained for the integral of hp from 0°C to the melting point. Data points were fit to an equation including a temperature-dependent, modified Loeb porosity correction.

#### *3.1.2 Equivalent conductivity and temperature drop across plenum gap*

Heat transfer coefficient hGg due to fission gas accumulating in the gap between the pellet and the sheath is a function of fission gas diffusing from the pellet towards the plenum gap. For fresh fuel, the conductivity is a function of helium thermal conductivity but the fission gas changes the gap conductivity. Change in the composition of the plenum gas is a function of fission gas accumulating in the plenum and it is estimated using the industry standard for estimation of the fraction of Xe and Kr [12] diffusing to the plenum as shown in **Table 2**.


**Table 1.**

*Correlation constants for different fuel types.*


**Table 2.**

*Temperature-dependent fraction of fission gas Xe and Kr in the plenum gap.*

The burnup-dependent yield of the fission product noble gases is input through a data file in FCCAL. To estimate the cumulative effect of fission gas in the plenum gap n-component gas mixture model is applied [13–15] as shown in Eq. (5).

$$\lambda\_{\text{mix}} = \sum\_{i=1}^{n} \frac{\lambda\_{\text{t}}}{1 + \sum\_{j=1}^{n} \mathbf{q}\_{ij} \frac{\mathbf{X}\_{i}}{\mathbf{X}\_{j}}} \tag{5}$$

Where, λmix and λ<sup>t</sup> are the thermal conductivities of the mixture gas and the individual component gases respectively, Xi and Xj are the mole fractions of the component gases and φij is constant. For the binary gas mixture consisting of He-Kr or He-Xe Eq. (1) may be written as:

$$
\lambda\_{\text{mix}} = \frac{\lambda\_1}{\mathbf{1} + \mathbf{q}\_{12}\frac{\mathbf{X}\_2}{\mathbf{X}\_1}} + \frac{\lambda\_2}{\mathbf{1} + \mathbf{q}\_{21}\frac{\mathbf{X}\_1}{\mathbf{X}\_2}} \tag{6}
$$

Where, subscript 1 is for heavier gas of the binary mixture. Values of φij [16] is shown in **Table 3**.

The values of φij for the component, mixture gases are independent of composition and temperature as shown by Gambhir and Saxena [17, 18] and are given by the following expression.

$$\frac{\text{\(\rho\_{\text{ij}} = \lambda\_{\text{i}}\)}\_{\text{\(\rho\_{\text{ji}} = \lambda\_{\text{j}}\)}} \frac{\lambda\_{\text{i}}}{\text{\(\mathbf{50M}}^2 + \mathbf{88M} + \mathbf{150}\)} \tag{7}$$

$$\mathbf{M} = \frac{\mathbf{M}\_2}{\mathbf{M}\_1} \tag{8}$$

These formulas are important because we can estimate the λmix of multicomponent gas mixtures if the thermal conductivity values of the corresponding binary and pure components are known. Moreover, these formulas help us to obtain λmix value at high temperature from knowledge of pure λ values at that temperature. Thus φijvalues determined at some lower temperature can be used to calculate λmix at some higher temperature.


**Table 3.**

*Mixture dependent constants for n (=2) component gas mixture equivalent conductivity formula.*
