Fuel Rods – Fuel Pins

Zirconium is a lustrous, grey-white, strong transition metal that resembles hafnium and, to a lesser extent, titanium. Zirconium is mainly used as a refractory and opacifier, although small amounts are used as an alloying agent for its strong resistance to corrosion. Zirconium alloy (e.g., Zr + 1%Nb) is widely used as a cladding for nuclear reactor fuels. The desired properties of these alloys are a low neutron-capture cross-section and resistance to corrosion under normal service conditions. Commercial non-nuclear grade zirconium typically contains 1–5% of hafnium, whose neutron absorption cross-section is 600x that of zirconium. Hafnium must therefore be almost entirely removed (reduced to < 0.02% of the alloy) for reactor applications. Zirconium alloys have lower thermal conductivity (about 18 W/m.K) than pure zirconium metal (about 22 W/m.K).

Fuel pellets are loaded and encapsulated within a fuel rod (a metallic cladding tube), which is made of zirconium alloys due to its very low absorption cross-section (unlike the stainless steel). Fuel rods are base element of a fuel assembly. Fuel rods have the purpose of containing fission products, ensuring mechanical support for the pellets, and allowing the heat removal to the coolant fluid of the heat generated by nuclear reactions. Typical fuel rod, has a length of some 4 m, with a diameter of around 1 cm. An 1100 MWe (3300 MWth) nuclear core may contain 157 fuel assemblies composed of over 45,000 fuel rods and some 15 million fuel pellets. Aside from the stacked pellets, and cladding tube, the rod comprises two welded endplugs, a plenum (or expansion chamber) to accommodate the fission gases released, and a spring, located inside the plenum, holding the fuel column in position. The fuel rod is filled with helium, at a pressure of about 25 bars, to compensate, in part, for the outside pressure, in the primary circuit (155 bars in PWRs).

Temperature Profile – Fuel Pellets

See also: Temperature Profile – Calculation

See also: Cladding Surface Temperature

Consider the fuel cladding of inner radius r_Zr,2 = 0.408 cm and outer radius r_Zr,1 = 0.465 cm. In comparison to fuel pellet, there is almost no heat generation in the fuel cladding (cladding is slightly heated by radiation). All heat generated in the fuel must be transferred via conduction through the cladding and therefore the inner surface is hotter than the outer surface.

To find the temperature distribution through the cladding we must solve the heat conduction equation. Due to symmetry in z-direction and in azimuthal direction, we can separate of variables and simplify this problem to one-dimensional problem. Thus, we will solve for the temperature as function of radius, T(r), only. In this example, we will assume that there is strictly no heat generation within the cladding. For constant thermal conductivity, k, the appropriate form of the cylindrical heat equation, is:

The general solution of this equation is:

where C₁ and C₂ are the constants of integration.

1)

Calculate the temperature distribution, T(r), in this fuel cladding, if:

• the temperature at the inner surface of the cladding is T_Zr,2 = 360°C
• the temperature of reactor coolant at this axial coordinate is T_bulk= 300°C
• the heat transfer coefficient (convection; turbulent flow) is h = 41 kW/m².K.
• the averaged material’s conductivity is k = 18 W/m.K
• the linear heat rate of the fuel is q_L = 300 W/cm and thus the volumetric heat rate is q_V = 597 x 10⁶ W/m³

From the basic relationship for heat transfer by convection we can calculate the outer surface of the cladding as:

As can be seen, also in this case we have given surface temperatures T_Zr,1 and T_Zr,2. This corresponds to the Dirichlet boundary condition. The constants may be evaluated using substitution into the general solution and are of the form:

Solving for C₁ and C₂ and substituting into the general solution, we then obtain:

∆T – cladding surface – coolant

Detailed knowledge of geometry, outer radius of cladding, linear heat rate, convective heat transfer coefficient and the coolant temperature determines ∆T between the coolant (T_bulk) and the cladding surface (T_Zr,1). Therefore we can calculate the cladding surface temperature (T_Zr,1) simply using the Newton’s Law:

∆T in fuel cladding

Detailed knowledge of geometry, outer and inner radius of cladding, linear heat rate, and the cladding surface temperature (T_Zr,1) determines ∆T between outer and inner surfaces of cladding. Therefore we can calculate the inner cladding surface temperature (T_Zr,2) simply using the Fourier’s Law:

• • 1. J. R. Lamarsh, Introduction to Nuclear Reactor Theory, 2nd ed., Addison-Wesley, Reading, MA (1983).
    2. J. R. Lamarsh, A. J. Baratta, Introduction to Nuclear Engineering, 3d ed., Prentice-Hall, 2001, ISBN: 0-201-82498-1.
    3. W. M. Stacey, Nuclear Reactor Physics, John Wiley & Sons, 2001, ISBN: 0- 471-39127-1.
    4. Glasstone, Sesonske. Nuclear Reactor Engineering: Reactor Systems Engineering, Springer; 4th edition, 1994, ISBN: 978-0412985317
    5. W.S.C. Williams. Nuclear and Particle Physics. Clarendon Press; 1 edition, 1991, ISBN: 978-0198520467
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    7. Robert Reed Burn, Introduction to Nuclear Reactor Operation, 1988.
    8. U.S. Department of Energy, Nuclear Physics and Reactor Theory. DOE Fundamentals Handbook, Volume 1 and 2. January 1993.

Advanced Reactor Physics:

• • 1. K. O. Ott, W. A. Bezella, Introductory Nuclear Reactor Statics, American Nuclear Society, Revised edition (1989), 1989, ISBN: 0-894-48033-2.
    2. K. O. Ott, R. J. Neuhold, Introductory Nuclear Reactor Dynamics, American Nuclear Society, 1985, ISBN: 0-894-48029-4.
    3. D. L. Hetrick, Dynamics of Nuclear Reactors, American Nuclear Society, 1993, ISBN: 0-894-48453-2. 
    4. E. E. Lewis, W. F. Miller, Computational Methods of Neutron Transport, American Nuclear Society, 1993, ISBN: 0-894-48452-4.