Fuel Performance Code – 1.5D Thermal analysis

In this section i propose to you a simple way to build up a nuclear fuel performance code which solve the heat equation considering the thermal expansion.

Domain and physical problem

The fuel pin in a nuclear reactor have a cylindrical shape and is made of three parts: the fuel itself, an inert gas and the cladding. Having a cylindrical shape mean that we can use cylindrical symmetry reducing the dimension from 3D to 2D (and also the radial domain considering only half of the fuel). A further approximation consist in decouple the axial from the radial problem reducing to a 1.5D description (in principle is not valid, due to swelling, cracking, hourglassing effects) we can subdivide the fuel in cylinders of smaller height and solve the radial heat equation in each one. To do so we have to know the boundary conditions: at the outer radius of the cladding we have Dirichlet BC, temperature is imposed by the coolant energy balance:

\Gamma c_p dT = q'dz

Knowing q’ we can simply find (by finite differences) the axial temperature profile of coolant and cladding (at outer radius):

T_{cladding,out}(z) = T_{coolant}(z)+\frac{q'(z)}{2\pi R_{cladding,out}h}

For what concern the left BC, we have that heat flux must be null due to symmetry, this mean Neumann BC, derivaty of the temperature equal to zero.

Now taht we know BCs we can focus on the heat equation: in this system the conductivity depend on the position (different materials) and on temperature, so the heat equation in steady state become:

\frac{1}{r} \left [\frac{dk}{dr}r\frac{dT}{dr} + k \left(\frac{dT}{dr} +r\frac{d^2T}{dr^2} \right) \right] + q''' = 0

Note that in general q”’ depends both from axial and radial coordinate, in this case i considered only the axial dependence, however it’s not difficult to impose also a radial profile.

Since the maximum temperature is the unknown and k depend on it, we have to perform an iterative procedure to calculate the conductivity: starting from a guess value for the temperature ranges in the various materials we solve the heat equation finding a first estimation of the temperature profile, from which we obtain a better description of the thermal conductivity k.

thermal conductivity in non expanded pin

Finally we can take into account the differential thermal expansion of fuel and cladding both in axial and radial direction, adopting again an iterative procedure: The idea is compute the temperature field in cold pin, from which we obtain the expanded geometry and then we repeat the calculation until convergence. doing so we have to consider that the expansion of the cladding affect the coolant velocity since the passage area now depend on the axial coordinate.

Thermal conductivity correlations

Oxide Fuel:

k_{unirradiated}(T,x,[Pu],p) = \left(\frac{1}{A_0 + A_x x + A_{Pu} [Pu] + (B_0 + B_{Pu}[Pu])T} + \frac{D}{T^2}e^{-\frac{E}{T}}\right)\left( 1 - p \right)^{2.5} \text{ } [W/mK]

k_{irradiated}(T,x,[Pu],p,bu) = k_{inf} + (k_0 - k_{inf})e^{-\frac{bu}{\phi}}

p is porosity, [Pu] the plutonium oxide concentration and x = 2 – O/M where O/M is the oxide to metal ratio. These correlations are valid for UO2 and MOX fuel.

Further informations about the fuel conductivity correlation, concerning the value of the coefficient and the validity range can be found in Magni article [1]:

Gap:

The typical correlation for rare gas (used in the gap between the fuel and the cladding):

k_{gas} = A \times 10^{-4}T^{0.79} \text{ } [W/mK]

In the case of Helium A = 15.8, see Todreas book [2] for a deeper explanation.

Code and algorithm

The code consist in three concatenated for cycle and the algorithm is explained below:

Main cycle 
{
    Compute the fluid velocity v(z) knowing the cladding radius R(z)
    Compute heat transfer coefficient h(z)
    Compute coolant temperature T_cool(z)
    Compute cladding outer temperature T_cl_out(z)

    Cycle over the z coordinate
    {
        Define radial domain and find the delimitation of the pin components
        Guess the temperature in the pin

        Cycle for convergence
        {
            Impose the temperature in the pin
            Compute the thermal conductivity k(T)
            Build the mathematical system and add BCs
            Solve the radial heat equation
            Update the temperature in the pin fot k(T)
        }

        Save the result
    }
    
    Compute thermal expansion R(z) and new height
    Create new axial domain
}

References

[1]

A. Magni, T. Barani, A. Del Nevo, D. Pizzocri, D. Staicu, P. Van Uffelen, L. Luzzi,
Modelling and assessment of thermal conductivity and melting behaviour of MOX fuel for fast reactor applications,
Journal of Nuclear Materials,
Volume 541,
2020,
152410,
ISSN 0022-3115,
https://doi.org/10.1016/j.jnucmat.2020.152410.

[2]

Neil E. Todreas, Mujid S. Kazimi,
Nuclear Systems volume 1: Thermal hydraulic fundamentals,
Second edition,
2011.