Prandtl number

${\displaystyle \mathrm{P}\mathrm{r}=\frac{\nu }{\alpha }=\frac{\text{momentum diffusivity}}{\text{thermal diffusivity}}=\frac{\mu /\rho }{k/(c_p\rho )}=\frac{c_p\mu }{k}}$

where:

• ${\displaystyle \nu }$ : momentum diffusivity (kinematic viscosity), ${\displaystyle \nu =\mu /\rho }$, (SI units: m²/s)
• ${\displaystyle \alpha }$ : thermal diffusivity, ${\displaystyle \alpha =k/(\rho c_p)}$, (SI units: m²/s)
• ${\displaystyle \mu }$ : dynamic viscosity, (SI units: Pa s = N s/m²)
• ${\displaystyle k}$ : thermal conductivity, (SI units: W/(m·K))
• ${\displaystyle c_p}$ : specific heat, (SI units: J/(kg·K))
• ${\displaystyle \rho }$ : density, (SI units: kg/m³).

Note that whereas the Reynolds number and Grashof number are subscripted with a scale variable, the Prandtl number contains no such length scale and is dependent only on the fluid and the fluid state. The Prandtl number is often found in property tables alongside other properties such as viscosity and thermal conductivity.

The mass transfer analog of the Prandtl number is the Schmidt number and the ratio of the Prandtl number and the Schmidt number is the Lewis number.

Experimental values

Typical values

For most gases over a wide range of temperature and pressure, Pr is approximately constant. Therefore, it can be used to determine the thermal conductivity of gases at high temperatures, where it is difficult to measure experimentally due to the formation of convection currents.

Typical values for Pr are:

• 0.003 for molten potassium at 975 K
• around 0.015 for mercury
• 0.065 for molten lithium at 975 K
• around 0.16–0.7 for mixtures of noble gases or noble gases with hydrogen
• 0.63 for oxygen
• around 0.71 for air and many other gases
• 1.38 for gaseous ammonia
• between 4 and 5 for R-12 refrigerant
• around 7.56 for water (At 18 °C)
• 13.4 and 7.2 for seawater (At 0 °C and 20 °C respectively)
• 50 for n-butanol
• between 100 and 40,000 for engine oil
• 1000 for glycerol
• 10,000 for polymer melts
• around 1×10²⁵ for Earth's mantle.

Formula for the calculation of the Prandtl number of air and water

For air with a pressure of 1 bar, the Prandtl numbers in the temperature range between −100 °C and +500 °C can be calculated using the formula given below. The temperature is to be used in the unit degree Celsius. The deviations are a maximum of 0.1% from the literature values.

${\displaystyle {\mathrm{P}\mathrm{r}}_{\text{air}}=\frac{{10}^9}{1.1\cdot {\vartheta }^3-1200\cdot {\vartheta }^2+322000\cdot \vartheta +1.393\cdot {10}^9}}$, where ${\displaystyle \vartheta }$ is the temperature in Celsius.

The Prandtl numbers for water (1 bar) can be determined in the temperature range between 0 °C and 90 °C using the formula given below. The temperature is to be used in the unit degree Celsius. The deviations are a maximum of 1% from the literature values.

${\displaystyle {\mathrm{P}\mathrm{r}}_{\text{water}}=\frac{50000}{{\vartheta }^2+155\cdot \vartheta +3700}}$