Conduction in fuel elements

Hana's second week begins at a whiteboard. Rohan draws a circle for a fuel pellet, a thin ring for the gap, a thicker ring for the sheath, and a wavy line for the coolant. "Every fuel limit in this business is a temperature somewhere on this drawing," he says. "The centre of the pellet must not melt, the gap must not grow so hot that fission gas floods out, and the sheath must stay cool enough to keep its strength and resist corrosion. Today you learn to walk across it."

Steady conduction in a cylindrical pellet

Consider a long cylindrical pellet of radius \(R\) with uniform volumetric heat generation \(q'''\) and conductivity \(k\). In steady state, with heat flowing only radially, the heat conduction equation in cylindrical coordinates is

Integrating once with the symmetry condition \(dT/dr = 0\) at the centre gives \(k\, dT/dr = -q''' r/2\). Integrating again from the centre to the surface, with constant \(k\), gives the temperature rise from surface to centre:

The second form is the key result of fuel element thermal design. Because \(q' = \pi R^{2} q'''\), the temperature rise across the pellet depends only on the linear heat rate and the conductivity. A fat pellet and a thin pellet at the same linear heat rate have the same centre-to-surface temperature difference. This is why fuel performance limits are written in kW/m and why a CANDU element at 50 kW/m has a much larger temperature drop than a PWR rod at 18 kW/m, whatever their diameters.

The conductivity of uranium dioxide

Uranium dioxide is chosen for its chemical stability, its high melting point of about 2800 °C, and its tolerance of radiation and fission products. Its weakness is its conductivity, which is low and falls with temperature. Unirradiated UO₂ has a conductivity of about 8 W/m·K at room temperature, about 3 W/m·K near 1000 °C and about 2 W/m·K near 1800 °C, before rising again slightly at very high temperatures as electronic conduction takes over. Burnup lowers it further, because fission products dissolve in the lattice and radiation damage scatters phonons; high-burnup fuel can have a conductivity a third lower than fresh fuel.

With temperature-dependent \(k\), the integration is done with the conductivity integral:

Fuel designers tabulate \(\int k\, dT\) from a reference temperature, so that the centre temperature can be read off directly once the surface temperature is known. The integral from 0 °C to melting is about 9 kW/m for UO₂. Setting \(q'/(4\pi)\) equal to the integral from the surface temperature to melting gives the linear heat rate at which the centre would just begin to melt, of order 60 to 70 kW/m for fresh fuel at typical surface temperatures. Design limits keep peak linear heat rates well below that value, with margin for uncertainties and transients.

The fuel-to-sheath gap

Fuel pellets are loaded into the sheath with a small radial gap, a few tens of micrometres, filled with helium. The gap allows the pellet to expand and swell without straining the sheath. It is also a thermal resistance, described by a gap conductance \(h_g\), so that the temperature drop across the gap is

Gap conductance has three parts: conduction through the gas, conduction through points where the pellet touches the sheath, and radiation, which is small except at very high temperatures. Typical values are about 5 to 10 kW/m²·K, but they change a lot during life. At the start, the gap is open and filled with helium, so conductance is modest. As the pellet cracks, relocates outward and swells, and as the sheath creeps inward under coolant pressure, the gap closes and contact conductance rises. Fission gases such as xenon and krypton released into the gap have much lower conductivity than helium, and they lower \(h_g\). This feedback, where higher temperatures release more gas, lower conductance and raise temperatures further, is one reason fuel performance codes are needed for accurate predictions.

CANDU fuel has a thinner sheath than light water reactor fuel, about 0.4 mm of Zircaloy-4, and is designed to collapse onto the pellets under the coolant pressure. The resulting good contact gives a high gap conductance. A thin graphite coating, called CANLUB, on the inside of the sheath reduces pellet-sheath interaction during power increases.

Sheath conduction and corrosion layers

The sheath is a thin cylindrical shell, so its temperature drop follows from the conduction equation without heat generation:

Zirconium alloys have a conductivity of about 15 to 18 W/m·K at operating temperature, so the drop is a few tens of degrees at high power. Over its life the sheath grows a layer of zirconium oxide on its outer surface. The oxide has a conductivity of only about 2 W/m·K, so even a few tens of micrometres add measurably to the temperature, and crud deposits from the coolant can add more. Waterside corrosion limits are therefore partly thermal limits.

Walking across the element

The full temperature profile is built up from the coolant outward. Starting with the bulk coolant temperature \(T_b\), the convective drop to the sheath surface, the sheath drop, the gap drop and the pellet drop are added in turn:

Every resistance is in series and every term is proportional to \(q'\), except that the pellet term must be found from the conductivity integral. The resistances are very unequal. The convective drop is typically 10 to 40 K, the sheath drop 20 to 50 K, the gap drop 100 to 300 K and the pellet drop several hundred to more than a thousand kelvin. The pellet itself is where most of the temperature difference lies.

# Radial temperature profile in a fuel rod with temperature-dependent UO2 conductivity.
import math
def k_uo2(T):            # W/m-K, simple fit for unirradiated UO2 (T in K), illustrative
    return 1 / (0.0375 + 2.165e-4 * T) + 4.715e9 / T**2 * math.exp(-16361 / T)

def pin(qp, T_cool, h, r_po, gap, t_clad, k_clad=17.0, h_gap=6000.0):
    """qp: linear heat rate W/m. Returns (T_clad_out, T_clad_in, T_fuel_surface, T_centre) in deg C."""
    r_ci = r_po + gap; r_co = r_ci + t_clad
    T_co = T_cool + qp / (2 * math.pi * r_co * h)
    T_ci = T_co + qp * math.log(r_co / r_ci) / (2 * math.pi * k_clad)
    T_fs = T_ci + qp / (2 * math.pi * r_po * h_gap)
    # conductivity integral: integral k dT from T_fs to T_c = q'/(4 pi)
    target, T, I = qp / (4 * math.pi), T_fs + 273.15, 0.0
    while I < target:
        I += k_uo2(T) * 1.0; T += 1.0
    return T_co, T_ci, T_fs, T - 273.15

cases = {
    'PWR rod at 18 kW/m':      (18e3, 310.0, 35000.0, 4.095e-3, 0.08e-3, 0.57e-3),
    'PWR hot rod at 40 kW/m':  (40e3, 320.0, 35000.0, 4.095e-3, 0.08e-3, 0.57e-3),
    'CANDU element at 50 kW/m': (50e3, 300.0, 40000.0, 6.08e-3, 0.04e-3, 0.42e-3),
}
for name, a in cases.items():
    co, ci, fs, c = pin(*a)
    print(f"{name:26s}: clad out {co:5.0f}, clad in {ci:5.0f}, pellet surface {fs:5.0f}, centre {c:5.0f} C")
print(f"q'/(4 pi) at 40 kW/m = {40e3/(4*math.pi):.0f} W/m (independent of pellet radius)")

PWR rod at 18 kW/m        : clad out   327, clad in   349, pellet surface   465, centre   797 C
PWR hot rod at 40 kW/m    : clad out   358, clad in   406, pellet surface   665, centre  1729 C
CANDU element at 50 kW/m  : clad out   330, clad in   361, pellet surface   580, centre  1902 C
q'/(4 pi) at 40 kW/m = 3183 W/m (independent of pellet radius)

Why centre temperature matters

Centre temperature is not just about melting. Fission gas release rises sharply above about 1200 to 1400 °C, as gas atoms diffuse to grain boundaries and escape into the gap. Released gas raises the internal pressure of the element and lowers gap conductance. Pellet swelling and thermal expansion, both larger at high temperatures, push on the sheath during power increases and can crack it by stress corrosion if the increase is too fast; this is called pellet-cladding interaction. Fuel operating rules therefore limit both the steady linear heat rate and the rate and size of power increases.

Centre temperature also sets the stored energy in the fuel, the heat that must be removed in an accident before cooling can catch up with decay heat. A pellet at 1700 °C at the centre and 700 °C at the surface holds far more energy than one at 800 °C and 500 °C. In a loss-of-coolant accident this stored energy flows into the sheath in the first few seconds, and it is a major contributor to the first peak in sheath temperature. Lower linear heat rates mean lower stored energy and larger accident margins.

Transient conduction

When power changes, the pellet temperature does not follow instantly. The thermal time constant of a fuel pellet is roughly

which for UO₂ pellets of a few millimetres radius is a few seconds. That delay is why fuel temperature feedback, the Doppler effect of NUC-300, acts within seconds but not instantly, and why short power spikes in reactivity transients deposit energy in the fuel faster than it can leave. Detailed transient fuel behaviour is calculated with time-dependent conduction solvers inside system codes and fuel performance codes, which Lesson 12 introduces.

For Hana, the CANDU numbers are a surprise. At 50 kW/m, with the coolant near 300 °C, the element centre runs near 1900 °C, hotter than the PWR hot rod, even though the coolant is cooler. Rohan points out that CANDU fuel is designed around this: natural uranium fuel reaches modest burnup, about 7500 to 9000 MW·d per tonne, and bundles leave the core within a year or two.

How fuel temperatures are measured and predicted

Nobody puts a thermocouple in a commercial fuel pellet. Fuel temperature models are built and checked in test reactors, where instrumented rods carry thermocouples at the pellet centre, pressure transducers in the plenum and extensometers on the stack. Programs at research reactors in several countries have produced decades of data on centre temperature, fission gas release and gap closure as functions of power and burnup. Fuel performance codes are fitted to and validated against those data. In Canada, the ELESTRES code models CANDU fuel elements in normal operation and ELOCA models them in accidents; light water reactor vendors and regulators use codes such as FRAPCON and FRAPTRAN in the United States and comparable codes elsewhere.

Commercial reactors provide indirect evidence. Post-irradiation examination in hot cells measures fission gas released from discharged elements, pellet grain growth that shows how hot the centre became, sheath strain and oxide thickness. Defect rates and the behaviour of fuel during power changes are tracked by utilities and vendors. Hana learns that each number she computes by hand has behind it a body of experiments, and that her job is to know which correlation came from which data and where it stops being trustworthy.