# Gap/Plenum Models

## Gap Heat Transfer

Gap heat transfer is modeled using the relation, (1) where is the total conductance across the gap, is the gas conductance, is the increased conductance due to solid-solid contact, and is the conductance due to radiant heat transfer.

The gas conductance is described using the form proposed by Ross and Stoute (1962): (2) where is the temperature dependent thermal conductivity of the gas in the gap, corresponds to the gap size (computed in the mechanics solution), is a roughness coefficient with and the roughnesses of the two surfaces, and and are jump distances at the two surfaces. The conductivity of the gas mixture () is computed based on the mixture rule from MATPRO (Allison et al., 1993), which permits mixtures of ten gases (helium, argon, krypton, xenon, hydrogen, nitrogen, oxygen, carbon monoxide, carbon dioxide, and water vapor). The gas temperature is the average of the local temperatures of the two surfaces. The value of takes the following form for `gap_geometry_type`

= PLATE, CYLINDER, and SPHERE respectively: (3) Temperature jump distance is calculated using Kennard's model based on a review by Lanning and Hann (1975). (4) where the units of , , and are , $ \frac{cal}{cm-K-s} is mole fraction of i-th gas species, is molcular weight of i-th gas species, and is accomodation coefficient for the gas mixture. The accomodation coefficients for helium and xenon are as follows: (5) (6) For a gas mixture, (7) where is molecular weight of xenon, is molecular weight of helium, and is molecular weight of gas mixture.

The increased conductance due to solid-solid contact, , is described using the correlation suggested by Ross and Stoute (1962): (8) where is an empirical constant, and are the thermal conductivities of the solid materials in contact, is the contact pressure, is the average gas film thickness (approximated as 0.8( + ), and is the Meyer hardness of the softer material. From measurements on steel in contact with aluminum, Ross and Stoute (1962) recommend = 10 , which is the default value in Bison. As an option, the chemical interaction layer at the fuel-cladding interface can be taken into account in the contact term. Based on experimental work (Kim, 2010), the growth of a (U,Zr)O layer is considered during fuel-cladding contact, and is described based on a parabolic law (9) where (m) is the layer thickness, and (Kim, 2010) is the parabolic growth rate. Equation Eq. 9 is solved numerically by (10) where is the layer thickness at the current time step (m), is the layer thickness at the previous time step (m), and is the time increment (s).

The chemical interaction layer is assumed to fill the fuel and cladding roughnesses according to its thickness, effectively reducing the and terms in Eq. 8 and improving the heat transfer.

The conductance due to radiant heat transfer, , is computed using a diffusion approximation. Based on the Stefan-Boltzmann law (11) where is the Stefan-Boltzmann constant, is an emissivity function, and and are the temperatures of the radiating surfaces. The radiant conductance is thus approximated (12) which can be reduced to (13) For infinite parallel plates, the emissivity function is defined as (14) where and are the emissivities of the radiating surfaces. This is the specific function implemented in Bison.

## Mechanical Contact

Mechanical contact between fuel pellets and the inside surface of the cladding is based on three requirements: (15) That is, the penetration distance (typically referred to as the gap in the contact literature) of one body into another must not be positive; the contact force opposing penetration must be positive in the normal direction; and either the penetration distance or the contact force must be zero at all times.

In Bison, these contact constraints are enforced through the use of node/face constraints. Specifically, the nodes of the fuel pellets are prevented from penetrating cladding faces. This is accomplished in a manner similar to that detailed by Heinstein and Laursen (1999). First, a geometric search determines which fuel pellet nodes have penetrated cladding faces. For those nodes, the internal force computed by the divergence of stress is moved to the appropriate cladding face at the point of contact. Those forces are distributed to cladding nodes by employing the finite element shape functions. Additionally, the pellet nodes are constrained to remain on the pellet faces, preventing penetration. Bison supports frictionless and tied contact. Friction is an important capability, and preliminary support for frictional contact is available.

Finite element contact is notoriously difficult to make efficient and robust in three dimensions. That being the case, effort is underway to improve the contact algorithm.

## Gap/plenum pressure

The pressure in the gap and plenum is computed based on the ideal gas law, (16) where is the gap/plenum pressure, is the moles of gas, is the ideal gas constant, is the temperature, and is the volume of the cavity. The moles of gas, the temperature, and the cavity volume in this equation are free to change with time. The moles of gas at any time is the original amount of gas (computed based on original pressure, temperature, and volume) plus the amount in the cavity due to fission gas released. The temperature is taken as the average temperature of the pellet exterior and cladding interior surfaces, though any other measure of temperature could be used. The cavity volume is computed as needed based on the evolving pellet and clad geometry.

## Gap/plenum temperature

The gap/plenum pressure (see above section) requires the temperature of the gas inside the cladding. Many choices are possible when supplying this temperature. It may be appropriate to supply the temperature at a node, the average temperature of several nodes, or data from an experiment. In this section, we outline an approach for calculating an average gas temperature that takes into account the entire fuel/cladding system.

We seek a weighted average temperature that accounts for the fact that the majority of the gas is in the plenum region. Using a volume-weighted average, the average gas temperature can be approximated as (17) where is the temperature at a point in the gap/plenum and is the volume occupied by the gas. It is necessary to make some approximations in the calculation of this temperature since the gap and plenum volumes are not meshed. We assume that a differential volume () is equal to a varying distance times a differential area (). This change is appropriate for replacing the integral over the volume of an enclosed space with the integral of the medial surface of that space times a distance representing the depth of the volume at a particular point on the surface.

With this change, it is necessary to replace with the temperature associated with . We take this temperature to be the average temperature of the outer and inner surfaces bounding the volume: (18) The medial surface of the gas volume is not known. We instead use the fuel surface. This gives (19) where is the fuel surface, is the temperature across the gap, is the temperature on the fuel surface, and is the gap distance. This approximation is a good one for the plenum region since the plenum volume can be accurately calculated given our assumptions. The accuracy of the calculation will be lower for the gap volume contribution, but since this volume is small (zero in areas of fuel/cladding contact) it is less important.

Note that since this approach places an appropriately large weight on the gas in the plenum, it is important that the temperature of the fuel adjacent to the plenum be accurate. It may be necessary to place insulating pellets in a model in order to calculate realistic temperatures at the top of the fuel stack.

## Integral Fuel Burnable Absorber (IFBA)

An integral fuel burnable absorber (IFBA) is used for optimizing fuel assembly reactivity and power distribution in a core. The IFBA is usually applied as a thin layer of over some length of a fuel rod. Since the IFBA layer is normally on the order of a few microns thick, the helium atoms generated are assumed to be released immediately into the plenum. In addition, the IFBA layer is depleted very quickly and is typically used up in the first of burnup or months of exposure.

Two models for the helium gas production (i.e., boron-10 depletion) have been implemented in Bison. These models are discussion on the IFBAHeProduction documentation page in detail.

## References

- C. M. Allison, G. A. Berna, R. Chambers, E. W. Coryell, K. L. Davis, D. L. Hagrman, D. T. Hagrman, N. L. Hampton, J. K. Hohorst, R. E. Mason, M. L. McComas, K. A. McNeil, R. L. Miller, C. S. Olsen, G. A. Reymann, and L. J. Siefken.
SCDAP/RELAP5/MOD3.1 code manual, volume IV: MATPROâ€“A library of materials properties for light-water-reactor accident analysis.
Technical Report NUREG/CR-6150, EGG-2720, Idaho National Engineering Laboratory, 1993.[BibTeX]
- M. Heinstein and T. Laursen.
An algorithm for the matrix-free solution of quasistatic frictional contact problems.
*International Journal for Numerical Methods in Engineering*, 44:1205â€“1226, 1999.[BibTeX] - K.-T. Kim.
UO$_2$/Zry-4 chemical interaction layers for intact and leak PWR fuel rods.
*Journal of Nuclear Materials*, 404():128â€“137, 2010.[BibTeX] - D. D. Lanning and C. R. Hann.
Review of methods applicable to the calculation of gap conductance in zircaloy-clad uo2 fuel rods.
Technical Report BWNL-1894, UC-78B, Pacific Northwest National Laboratory, 1975.[BibTeX]
- A. M. Ross and R. L. Stoute.
Heat transfer coefficient between UO$_2$ and Zircaloy-2.
Technical Report AECL-1552, Atomic Energy of Canada Limited, 1962.[BibTeX]