Prandtl number

Physical key figure
Surname	Prandtl number
Formula symbol	${\displaystyle {\mathit {Pr}}}$
dimension	dimensionless
definition	${\displaystyle {\mathit {Pr}}={\frac {\nu }{a}}}$
${\displaystyle \nu }$ kinematic viscosity ${\displaystyle a}$ Thermal diffusivity
Named after	Ludwig Prandtl
scope of application	Comparison of convection and diffusion

The Prandtl number ( ) is a dimensionless number of fluids named after Ludwig Prandtl , i.e. gases or liquids that can drip. It is defined as the relationship between kinematic viscosity and thermal diffusivity : ${\displaystyle {\mathit {Pr}}}$

${\displaystyle {\mathit {Pr}}={\frac {\nu }{a}}={\frac {\nu \,\rho \,c_{\mathrm {p} }}{\lambda }}={\frac {\eta \,c_{\mathrm {p} }}{\lambda }}}$

• ${\displaystyle \eta }$- dynamic viscosity of the fluid in kg m ⁻¹ s ⁻¹
• ${\displaystyle \nu }$- kinematic viscosity in m ² s ⁻¹
• ${\displaystyle \lambda }$- Thermal conductivity in W m ⁻¹ K ⁻¹
• ${\displaystyle a}$- Thermal diffusivity in m ² · s ⁻¹
• ${\displaystyle c_{\mathrm {p} }}$- specific heat capacity in J kg ⁻¹ K ⁻¹ at constant pressure.

The Prandtl number represents the link between the velocity field and the temperature field of a fluid. While the kinematic viscosity represents the impulse transport as a result of friction, the thermal conductivity coefficient stands for the (possibly unsteady) heat transport as a result of conduction. Since the momentum transport is determined by the velocity field and the heat transport is determined by the temperature field, the Prandtl number connects the two fields that are decisive for heat transfer. The Prandtl number is therefore a measure of the ratio of the thicknesses of the flow boundary layer to the temperature boundary layer . ${\displaystyle \nu }$ ${\displaystyle a}$

The Prandtl number is a pure, temperature- and pressure-dependent in general fabric size (material parameter) of the fluid: . ${\displaystyle {\mathit {Pr}}={\mathit {Pr}}(T,p)}$

The analogue of the Prandtl number in mass transfer is the Schmidt number . The ratio of Schmidt and Prandtl numbers is the Lewis number . ${\displaystyle {\mathit {Sc}}}$

For a model gas consisting of uniform, hard spheres with an attractive dipole interaction ( hard spherical gas), the value is obtained regardless of the temperature (see kinetic gas theory ). This stands for monatomic gases helium , neon , argon , krypton and xenon in very good agreement with the experimental values. ${\displaystyle Pr={\tfrac {2}{3}}}$

For gases and vapors, the following applies approximately to pressures from 0.1 to 10 bar: ${\displaystyle {\mathit {Pr}}={\frac {4\kappa }{9\kappa -5}}}$

where is the isentropic exponent . ${\displaystyle \kappa }$

Prandtl numbers of important heat transfer media

Prandtl number of air as a function of temperature and pressure

Gases material Prandtl number air 00.7179 (0 ° C, 1 bar abs) 00.7194 (500 ° C, 1 bar abs) Steam 00.973 (100 ° C) 00.869 (500 ° C)

liquids material Prandtl number water 13.44 (0 ° C) 11.16 (5 ° C) 06.99 (20 ° C) 04.34 (40 ° C) 03.00 (60 ° C) 02.20 (80 ° C) 01, 75 (100 ° C) Sodium Note 1 00000.0114 (100 ° C) 00000.00535 (350 ° C) mercury 00000.0232 benzene 00007.488 Ethanol 00018.84 Ethylene glycol 00184.6 Glycerin 11340

^{Note 1} Melting point: 97.72 ° C

In general:

• The Prandtl numbers of liquids decrease with increasing temperature.
• Liquid metals have very small Prandtl numbers.

Usage formulas for air and water

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

${\displaystyle Pr_{\text{Luft}}={\frac {10^{9}}{1,1\cdot \vartheta ^{3}-1200\cdot \vartheta ^{2}+322000\cdot \vartheta +1,393\cdot 10^{9}}}}$

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 entered in degrees Celsius. The deviations are a maximum of 1% compared to the literature values.

${\displaystyle Pr_{\text{Wasser}}={\frac {50000}{\vartheta ^{2}+155\cdot \vartheta +3700}}}$

Prandtl number in turbulent flows

In the case of turbulent flows, the strong turbulence caused increased diffusivity:

${\displaystyle \nu _{\text{total}}={\nu }+{\nu _{\mathrm {t} }}}$

${\displaystyle a_{\text{total}}={a}+{a_{\mathrm {t} }}}$

This also allows a turbulent Prandtl number to be defined:

${\displaystyle {\mathit {Pr_{\mathrm {t} }}}={\frac {\nu _{\mathrm {t} }}{a_{\mathrm {t} }}}}$

The turbulent Prandtl number is useful for calculating turbulent boundary layer flows with heat transfer. In the simple model of the Reynolds analogy is . Experimental data for air currents give a more accurate value of 0.7-0.9. ${\displaystyle {Pr_{\mathrm {t} }}=1}$

Individual evidence

1. ↑ H. Brauer: Exchange of substances including chemical reactions. Sauerländer AG, Aarau, 1971, ISBN 3794100085
2. ↑ Prandtl numbers for liquids (PDF; 248 kB)
3. ↑ tec-science: Prandtl number. In: tec-science. May 9, 2020, accessed on June 25, 2020 (German). 
4. ↑ tec-science: Prandtl number. In: tec-science. May 9, 2020, accessed on June 25, 2020 (German).