## Prandtl Number of Liquid Metals

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.

The **Prandtl number** enters many calculations of heat transfer in liquid metal reactors. Two of promising designs of Generation IV reactors use a liquid metal as the reactor coolant. The development of Generation IV reactors represents a challenge from an engineering point of view.

- Sodium-cooled fast reactor
- Lead-cooled fast reactor

One of the main challenges in numerical simulation is the reliable modeling of heat transfer in liquid-metal cooled reactors by Computational Fluid Dynamics (CFD). Heat transfer applications with **low-Prandtl number fluids are** often in the transition range between conduction and convection dominated regimes.

## Prandtl Number

The **Prandtl number** is a dimensionless number, named after its inventor, a German engineer **Ludwig Prandtl**, who also identified the boundary layer. The** Prandtl number** is defined as the** ratio** of **momentum diffusivity** to **thermal diffusivity**. The **momentum diffusivity**, or as it is normally called, kinematic viscosity, tells us the material’s resistance to shear-flows (different layers of the flow travel with different velocities due to e.g. different speeds of adjacent walls) in relation to density. That is, the **Prandtl number** is given as:

where:

**ν** is **momentum diffusivity** (kinematic viscosity) [m^{2}/s]

**α** is **thermal diffusivity** [m^{2}/s]

**μ** is **dynamic viscosity** [N.s/m^{2}]

**k** is **thermal conductivity** [W/m.K]

**c _{p }**is

**specific heat**[J/kg.K]

**ρ** is **density** [kg/m^{3}]

Small values of the **Prandtl number**, **Pr << 1**, means the thermal diffusivity dominates. Whereas with large values, **Pr >> 1**, the momentum diffusivity dominates the behavior. For example, the typical value for liquid mercury, which is about 0.025, indicates that the **heat conduction** is more significant compared to **convection**, so thermal diffusivity is dominant. When Pr is small, it means that the heat diffuses quickly compared to the velocity.

In comparison to Reynolds number, the **Prandtl number** is not dependent on geometry of an object involved in the problem, but is dependent solely on the fluid and the fluid state. As such, the **Prandtl number** is often found in property tables alongside other properties such as viscosity and thermal conductivity.

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