Nusselt Number

The Nusselt number is a dimensionless number, named after a German engineer Wilhelm Nusselt. The Nusselt number is closely related to Péclet number and both numbers are used to describe the ratio of the thermal energy convected to the fluid to the thermal energy conducted within the fluid. Nusselt number is equal to the dimensionless temperature gradient at the surface, and it provides a measure of the convection heat transfer occurring at the surface. The conductive component is measured under the same conditions as the heat convection but with a stagnant fluid. The Nusselt number is to the thermal boundary layer what the friction coefficient is to the velocity boundary layer. Thus, the Nusselt number is defined as:

Nusselt Number - definition

where:

kf is thermal conductivity of the fluid [W/m.K]

L is the characteristic length

h is the convective heat transfer coefficient [W/m2.K]

For illustration, consider a fluid layer of thickness L and temperature difference ΔT. Heat transfer through the fluid layer will be by convection when the fluid involves some motion and by conduction when the fluid layer is motionless.

In case of conduction, the heat flux can be calculated using Fourier’s law of conduction. In case of convection, the heat flux can be calculated using Newton’s law of cooling. Taking their ratio gives:

nusselt number - convection to conduction

The preceding equation defines the Nusselt number. Therefore, the Nusselt number represents the enhancement of heat transfer through a fluid layer as a result of convection relative to conduction across the same fluid layer.  A Nusselt number of Nu=1 for a fluid layer represents heat transfer across the layer by pure conduction. The larger the Nusselt number, the more effective the convection. A larger Nusselt number corresponds to more effective convection, with turbulent flow typically in the 100–1000 range. For turbulent flow, the Nusselt number is usually a function of the Reynolds number and the Prandtl number.

Correlations – Single-phase Fluid Flow

The average Nusselt number over the entire plate is determined by:

laminar flow - flat plate - nusselt number

This relation gives the average heat transfer coefficient for the entire plate when the flow is laminar over the entire plate.

Constant Surface Temperature

In laminar flow in a tube with constant surface temperature, both the friction factor and the heat transfer coefficient remain constant in the fully developed region.

Laminar Flow - Circular Tube - temperature

Constant Surface Heat Flux

Therefore, for fully developed laminar flow in a circular tube subjected to constant surface heat flux, the Nusselt number is a constant. There is no dependence on the Reynolds or the Prandtl numbers.

Laminar Flow - Circular Tube - flux

The average Nusselt number over the entire plate is determined by:

turbulent flow - flat plate - nusselt number

This relation gives the average heat transfer coefficient for the entire plate only when the flow is turbulent over the entire plate, or when the laminar flow region of the plate is too small relative to the turbulent flow region.

See also: Dittus-Boelter Equation

For fully developed (hydrodynamically and thermally) turbulent flow in a smooth circular tube, the local Nusselt number may be obtained from the well-known Dittus-Boelter equation. The Dittus–Boelter equation is easy to solve but is less accurate when there is a large temperature difference across the fluid and is less accurate for rough tubes (many commercial applications), since it is tailored to smooth tubes.

Dittus-Boelter Equation - Formula

The Dittus-Boelter correlation may be used for small to moderate temperature differences, Twall – Tavg, with all properties evaluated at an averaged temperature Tavg.

For flows characterized by large property variations, the corrections (e.g. a viscosity correction factor μ/μwall) must be taken into account, for example, as Sieder and Tate recommend.

Nusselt Number for Liquid Metal Reactors

See also: Nusselt Number for Liquid Metal Reactors

For liquid metals the Prandtl number is very small, generally in the range from 0.01 to 0.001. This means that the thermal diffusivity, which is related to the rate of heat transfer by conduction, unambiguously dominates. This very high thermal diffusivity results from very high thermal conductivity of metals, which is about 100 times higher than that of water. The Prandtl number for sodium at a typical operating temperature in the Sodium-cooled fast reactors is about 0.004. For this case the thermal boundary layer development is much more rapid than that of the velocity boundary layer (δt >> δ), and it is reasonable to assume uniform velocity throughout the thermal boundary layer.

Heat transfer coefficients for sodium flow through fuel channel are based on the Prandtl number and Péclet number. Pitch-to-diameter (P/D) also enters many calculations of heat transfer in liquid metal reactors. Convective heat transfer correlations are usually presented in terms of Nusselt number versus Péclet number. Typical Péclet number for normal operation are from 150 to 300 in the fuel bundles. As for another flow regimes,  the Nusselt number and a given correlation can be used to determine the convective heat transfer coefficient.

Example – Nusselt Number – Cladding Surface Temperature

Convection - Convective Heat TransferCladding 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, therefore its coolability is one of key safety aspects.

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.

Assume that:

  • the outer diameter of the cladding is: d = 2 x rZr,1 = 9,3 mm
  • the pitch of fuel pins is: p = 13 mm
  • the thermal conductivity of saturated water at 300°C is: kH2O = 0.545 W/m.K
  • the dynamic viscosity of saturated water at 300°C is: μ = 0.0000859 N.s/m2
  • the fluid density is: ρ = 714 kg/m3
  • the specific heat is: cp = 5.65 kJ/kg.K
  • the core flow velocity is constant and equal to Vcore = 5 m/s
  • the temperature of reactor coolant at this axial coordinate is: Tbulk = 296°C
  • the linear heat rate of the fuel is qL = 300 W/cm (FQ ≈ 2.0) and thus the volumetric heat rate is qV = 597 x 106 W/m3

Hydraulic Diameter - Fuel ChannelCalculate the Prandtl, Reynolds and Nusselt number for this flow regime (internal forced turbulent flow) inside the rectangular fuel lattice (fuel channel), then calculate the heat transfer coefficient and finally the cladding surface temperature, TZr,1.

To calculate the cladding surface temperature, we have to calculate the Prandtl, Reynolds and Nusselt number, because the heat transfer for this flow regime can be described by the Dittus-Boelter equation, which is:

Dittus-Boelter Equation - Formula

Calculation of the Prandtl number

To calculate the Prandtl number, we have to know:

  • the thermal conductivity of saturated water at 300°C is: kH2O = 0.545 W/m.K
  • the dynamic viscosity of saturated water at 300°C is: μ = 0.0000859 N.s/m2
  • the specific heat is: cp = 5.65 kJ/kg.K

Note that, all these parameters significantly differs for water at 300°C from those at 20°C.  Prandtl number for water at 20°C is around 6.91. Prandtl number for reactor coolant at 300°C is then:

prandtl number - example

Calculation of the Reynolds number

To calculate the Reynolds number, we have to know:

  • the outer diameter of the cladding is: d = 2 x rZr,1 = 9,3 mm (to calculate the hydraulic diameter)
  • the pitch of fuel pins is: p = 13 mm  (to calculate the hydraulic diameter)
  • the dynamic viscosity of saturated water at 300°C is: μ = 0.0000859 N.s/m2
  • the fluid density is: ρ = 714 kg/m3

The hydraulic diameter, Dh, is a commonly used term when handling flow in non-circular tubes and channels. The hydraulic diameter of the fuel channel, Dh, is equal to 13,85 mm.

See also: Hydraulic Diameter

The Reynolds number inside the fuel channel is then equal to:

reynolds number - example

This fully satisfies the turbulent conditions.

Calculation of the Nusselt number using Dittus-Boelter equation

For fully developed (hydrodynamically and thermally) turbulent flow in a smooth circular tube, the local Nusselt number may be obtained from the well-known Dittus–Boelter equation.

To calculate the Nusselt number, we have to know:

The Nusselt number for the forced convection inside the fuel channel is then equal to:

nusselt number - example

Calculation of the heat transfer coefficient and the cladding surface temperature, TZr,1

Detailed knowledge of geometry, fluid parameters, outer radius of cladding, linear heat rate, convective heat transfer coefficient allows us to calculate the temperature difference ∆T between the coolant (Tbulk) and the cladding surface (TZr,1).

To calculate the the cladding surface temperature, we have to know:

  • the outer diameter of the cladding is: d = 2 x rZr,1 = 9,3 mm
  • the Nusselt number, which is NuDh = 890
  • the hydraulic diameter of the fuel channel is: Dh = 13,85 mm
  • the thermal conductivity of reactor coolant (300°C) is: kH2O = 0.545 W/m.K
  • the bulk temperature of reactor coolant at this axial coordinate is: Tbulk = 296°C
  • the linear heat rate of the fuel is: qL = 300 W/cm (FQ ≈ 2.0)

The convective heat transfer coefficient, h, is given directly by the definition of Nusselt number:

convective heat transfer coefficient - example

Finally, we can calculate the cladding surface temperature (TZr,1) simply using the Newton’s Law of Cooling:

Newton law of cooling - example

For PWRs at normal operation, there is a compressed liquid water inside the reactor core, loops and steam generators.  The pressure is maintained at approximately 16MPa. At this pressure water boils at approximately 350°C(662°F). As can be seen, the surface temperature TZr,1 = 325°C ensures, that even subcooled boiling does not occur. Note that, subcooled boiling requires TZr,1 = Tsat. Since the inlet temperatures of the water are usually about 290°C (554°F), it is obvious this example corresponds to the lower part of the core. At higher elevations of the core the bulk temperature may reach up to 330°C. The temperature difference of 29°C causes the subcooled boiling may occur (330°C + 29°C > 350°C). On the other hand, nucleate boiling at the surface effectively disrupts the stagnant layer and therefore nucleate boiling significantly increases the ability of a surface to transfer thermal energy to bulk fluid. As a result, the convective heat transfer coefficient significantly increases and therefore at higher elevations, the temperature difference (TZr,1 – Tbulk) significantly decreases.

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: