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Thermal Conductivity Formula – Equation

Example - Heat Flux - Thermal ConductionThe heat transfer characteristics of solid material are measured by a property called the thermal conductivity, k (or λ), measured in W/m.K. It measures a substance’s ability to transfer heat through a material by conduction. Note that Fourier’s law applies to all matter, regardless of its state (solid, liquid, or gas). Therefore, it is also defined for liquids and gases.

The thermal conductivity of most liquids and solids varies with temperature, and for vapors, it also depends upon pressure. In general:

thermal conductivity - definition

Most materials are nearly homogeneous. Therefore, we can usually write k = k (T). Similar definitions are associated with thermal conductivities in the y- and z-directions (ky, kz), but for an isotropic material, the thermal conductivity is independent of the direction of transfer, kx = ky = kz = k.

Fourier’s law is an expression that defines thermal conductivity.

Fourier’s law of Thermal Conduction

Uranium Dioxide – Thermal Conductivity Formula

Thermal conduction - thermal conductivity - uranium dioxideMost PWRs use uranium fuel, which is in the form of uranium dioxide. Uranium dioxide is a black semiconducting solid with very low thermal conductivity. On the other hand, uranium dioxide has a very high melting point and has well-known behavior. The UO2 is pressed into pellets, and these pellets are then sintered into the solid.

These pellets are then loaded and encapsulated within a fuel rod (or fuel pin) made of zirconium alloys due to their very low absorption cross-section (unlike stainless steel). The surface of the tube, which covers the pellets, is called fuel cladding, and fuel rods are the base element of a fuel assembly.

The thermal conductivity of uranium dioxide is very low compared with metal uranium, uranium nitride, uranium carbide, and zirconium cladding material. Thermal conductivity is one parameter that determines the fuel centerline temperature. This low thermal conductivity can result in localized overheating in the fuel centerline, and therefore this overheating must be avoided. Overheating of the fuel is prevented by maintaining the steady-state peak linear heat rate (LHR) or the Heat Flux Hot Channel Factor – FQ(z) below the level at which fuel centerline melting occurs. Expansion of the fuel pellet upon centerline melting may cause the pellet to stress the cladding to the point of failure.

The thermal conductivity formula of solid UO2 with a density of 95% is estimated by the following correlation [Klimenko; Zorin]:

thermal conductivity of uranium - equation

where τ = T/1000. The uncertainty of this correlation is +10% in the range from 298.15 to 2000 K and +20% in the range from 2000 to 3120 K.

Thermal Conductivity - Uranium Dioxide - chart

Special reference: Thermal and Nuclear Power Plants/Handbook ed. by A.V. Klimenko and V.M. Zorin. MEI Press, 2003.

Special reference: Thermophysical Properties of Materials For Nuclear Engineering: A Tutorial and Collection of Data. IAEA-THPH, IAEA, Vienna, 2008. ISBN 978–92–0–106508–7.

 
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.
  3. Fundamentals of Heat and Mass Transfer. C. P. Kothandaraman. New Age International, 2006, ISBN: 9788122417722.
  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.
  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
  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.

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.

See above:

Thermal Conductivity