This type of modeling using an FEM program has been per- formed previously [11]。 The modeled results for the thermal conductivity were compared to experimentally measured val- ues for a composite of molybdenum fibers dispersed in alumina。 The difference between that modeling method and the one in the present study involves the way in which the possible overlapping of randomly inserted fibers is treated。

Xu [11] used the length of the fiber as the minimum distance the centers of two fibers may approach each other, thus not al- lowing the fibers to be close to each other。 The program created here allows for fibers to be closer than this distance, but uses an orientation correction to prevent fibers from overlapping。 In- tuitively, this gives a more physically realistic representation of random fibers, as more placement possibilities exist。

The geometry of the whisker composite was produced in a Microsoft Excel program。 The basis of the program was to produce random whisker positions and orientations in a UO2 matrix。

This modeling is being performed in a 2-D plane as the sim- plest geometry setup, as shown in Figure 6, for one random ar- rangement。 For reasons that are not clear, an apparent clustering of whiskers occurs for these rather short, fat whiskers。

The geometry program uses a random number generator to create random whisker positions and orientations。 Then sim- ple logic arguments are used to eliminate the occurrence of whisker overlap。 The program first lays down a grid with spac- ing consisting of the minimum square dimension of a whisker, which is the square of the whisker’s  width。  Then, random grid point locations and whisker orientations are produced。 A whisker’s position is cross-checked with all other whiskers to determine if it is in the minimum radius, the whisker’s length, where interactions might occur。 If another whisker is inside of this radius, then the distance between the two center points of the whiskers is calculated。 The offending whisker is deter- mined to be in one of four quadrants in relation to the whisker designated to be at the origin。 The information of  the dis- tance between whiskers and the relative position from one an- other is used to determine the orientation correction to prevent overlap。

Figure 6 The FEM thermal model geometries for the UO2/SiC whisker fuels with (a) 2。5 vol %, and (b) 5。0 vol % of SiC whiskers。

The orientation correction was created from a curve fit。 The basis for this is that the greatest interaction occurs when two whiskers are parallel and approach each other along a line。 Therefore, one whisker must be offset by some angle to prevent overlap as the two whiskers approach one another。 The solution is a polynomial curve fit that relates the whisker’s width and dis- tance from one another to the minimum angle that one whisker must be offset in order to prevent overlapping。 The results of this program produce acceptable geometries that have little to no whisker interactions when executed。

The final result of the geometry program is a square ma- trix of UO2 with embedded SIC whiskers。 The geometry was then entered into ANSYS, and the thermal conductivity was calculated following the same procedure as previous thermal modeling work。 Some valid concerns from Xu [11] were ad- dressed, such as the suggested optimum number of fibers being thirty。 They had determined this optimum number by adding fibers to their thermal model until the error associated with the bulk orientation of the block came to a minimum at thirty fibers。 Adding fibers beyond this number did not improve the unifor- mity of the thermal conductivity calculation with respect to bulk orientation。 The first geometry was set up with thirty whiskers, and the thermal conductivity was calculated with a heat flux applied along the X axis and then along the Y axis。 The calcu- lations determined that the differences in thermal conductivity between the two directions were on average six percent。 This number is consistent with differences seen in Xu [9]。 Therefore, when a thermal conductivity is reported from this modeling, it will be an average between the two values for an applied heat flux。