# Temperature distribution in Fuel Cladding

## Temperature distribution in Fuel Cladding

Cladding is the outer layer of the fuel rods, standing between the reactor coolant and the nuclear fuel (i.e. fuel pellets). It is made of a corrosion-resistant material with low absorption cross section for thermal neutrons, usually zirconium alloy. Cladding prevents radioactive fission products from escaping the fuel matrix into the reactor coolant and contaminating it. Cladding constitute one of barriers in ‘defence-in-depth‘ approach.

Consider the fuel cladding of inner radius rZr,2 = 0.408 cm and outer radius rZr,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 TZr,2 = 360°C
• the temperature of reactor coolant at this axial coordinate is Tbulk = 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 qL = 300 W/cm and thus the volumetric heat rate is qV = 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 TZr,1 and TZr,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 (Tbulk) and the cladding surface (TZr,1). Therefore we can calculate the cladding surface temperature (TZr,1) simply using the Newton’s Law:

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

References:
Heat Transfer:
1. Fundamentals of Heat and Mass Transfer, 7th Edition. Theodore L. Bergman, Adrienne S. Lavine, Frank P. Incropera. John Wiley & Sons, Incorporated, 2011. ISBN: 9781118137253.
2. Heat and Mass Transfer. Yunus A. Cengel. McGraw-Hill Education, 2011. ISBN: 9780071077866.
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4. U.S. Department of Energy, Thermodynamics, Heat Transfer and Fluid Flow. DOE Fundamentals Handbook, Volume 2 of 3. May 2016.

Nuclear and Reactor Physics:

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.
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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
6. G.R.Keepin. Physics of Nuclear Kinetics. Addison-Wesley Pub. Co; 1st edition, 1965
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.
9. Paul Reuss, Neutron Physics. EDP Sciences, 2008. ISBN: 978-2759800414.