# Layered1D (1.5D) Capability Introduction

Layered 1D models, sometimes refered to as 1.5D, consist of several 1D cylinderical fuel 'slices' that are coupled together with out-of-plane strain variables. These models, which are composed of fuel slices at specific axial positions for the duration of the simulation, are best suited to model problems primarily concerned with radial heat transfer.

The **Layered 1D Tutorial** provides step-by-step instructions for creating the input file for one of the Bison Layered 1D example problems.

## Introduction to Layered 1D Models

The term 'Layered1D' in Bison describes a model of cylindrical fuel geometry that is a collection of coupled 1D simulations used to represent the behavior of the fuel at given axial positions along the rod. The fuel rod is divided into several axial slices, and a 1D model of the physics, Eq. 1 and Eq. 2, is solved on each slice. An illustration of these 1D axial slices is shown in Figure 1.

The following equations for energy conservation and stress divergence, respectively, are solved simultaneously using 1D axisymmetric models for each axial slice: (1) (2) where is the mass density, is the specific heat, is the temperature, is the thermal conductivity, is the volumetric heat generation rate, and is the stress.

Each slice represents the behavior of a finite axial segment of the rod. The behavior is assumed to be uniform over the entirety each axial segement.

## Layered 1D Specific Code

Because a given 1D model is only applicable for a finite length of the fuel rod, and generally the behavior of a rod varies significantly over its length, the ability to represent axial variations is necessary. This ability is added to Bison with a set of code classes that compute integral quantities specifically for Layered 1D models, including code used to calculate:

Plenum volume (InternalVolumeLayered1D)

Fission gas released (SideAverageValueLayered1D computes the average temperature used to determine fission gas released)

Heat flux (SideFluxIntegralLayered1D)

Burnup profiles (Layered1DFuelPinGeometry provides geometry information about the fuel rod geometry)

Axial pressure contributions from the coolant and plenum pressure (

`out_of_plane_pressure`

parameter in Layered1DMaster)

These Layered1D specific code classes are aware that the domain is composed of a set of 1D models with finite axial length, and compute the integrals in an appropriate manner. All of these code classes specific to Layered 1D models follow the nomenclature convention of using **Layered1D** as a suffix for the class name.

## Coupling with Axial Out-of-Plane Strains

A out-of-plane strain condition must be imposed on the mechanical response of the 1D axial slices to couple the individual slices into a Layered 1D simulation. In continum mechanics, plane strain is defined with the aussumption that out-of-plane strains are zero. However, in the case of a fuel rod, the axial strains are generally required to be nonzero in order to capture the effects of thermal explansion and other phenomena leading to volume change. At the same time, these axial strains must be uniform to ensure that the planar sections of the axial slices remain planar.

A generalized plane strain capability solves for the out-of-plane strain that satisfies equilibrium in this case. Read more about this capability in the MOOSE Generalized Plane Strain documentation. In the Layered1D fuel rods, an independent generalized plane strain condition is imposed on the fuel and cladding at every axial slice position.

## Comparison with 2D-RZ Simulations

For regular fuel rods under normal operating conditions, a Layered1D simulation can replicate the temperature and radial displacements from a 2D-RZ simulation. Because of the simplifying assumptions made about the axial and azmuthial displacements and strains in a layered1D simulation, the strains in a layered1D simulation will follow the same trends as those in a 2D-RZ simulation but will not match. These results and conclusions are detailed in the CASL Verify and Validate 1.5D Capability milestone report (Pitts et al., 2017).

## Best Use Cases for Layered 1D Models

The Layered 1D capability has been developed to address the need for a fast-running, simplified version of Bison for use in large multi-physics simulations involving hundreds of individual fuel rods in VERA. Layered 1D models cannot capture localized effects, ranging from manufacturing defects to LOCA conditions, which require higher dimension fidelity. The Bison Layered 1D capability is ideal for simulations primarily focused on capturing radial heat transfer under normal operating conditions.

## References

- S. A. Pitts, S. R. Novascone, H. Chen, B. W. Spencer, S. Satpathy, R. J. Gardner, and J. D. Hales.
Verify and validate 1.5d capability.
Technical Report CASL-U-2017-1380-000, Consortium for Advanced Simulation of LWRs, 2017.[BibTeX]