The Prandtl number (Pr) or Prandtl group is a
dimensionless number, named after the German physicist
Ludwig Prandtl, defined as the ratio of
momentum diffusivity to
thermal diffusivity
In heat transfer analysis, thermal diffusivity is the thermal conductivity divided by density and specific heat capacity at constant pressure. It measures the rate of transfer of heat of a material from the hot end to the cold end. It has the SI ...
.
The Prandtl number is given as:
:
where:
*
: momentum diffusivity (
kinematic viscosity),
, (
SI units: m
2/s)
*
:
thermal diffusivity
In heat transfer analysis, thermal diffusivity is the thermal conductivity divided by density and specific heat capacity at constant pressure. It measures the rate of transfer of heat of a material from the hot end to the cold end. It has the SI ...
,
, (SI units: m
2/s)
*
:
dynamic viscosity, (SI units: Pa s = N s/m
2)
*
:
thermal conductivity
The thermal conductivity of a material is a measure of its ability to conduct heat. It is commonly denoted by k, \lambda, or \kappa.
Heat transfer occurs at a lower rate in materials of low thermal conductivity than in materials of high thermal ...
, (SI units: W/(m·K))
*
:
specific heat
In thermodynamics, the specific heat capacity (symbol ) of a substance is the heat capacity of a sample of the substance divided by the mass of the sample, also sometimes referred to as massic heat capacity. Informally, it is the amount of heat t ...
, (SI units: J/(kg·K))
*
:
density
Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematicall ...
, (SI units: kg/m
3).
Note that whereas the
Reynolds number
In fluid mechanics, the Reynolds number () is a dimensionless quantity that helps predict fluid flow patterns in different situations by measuring the ratio between inertial and viscous forces. At low Reynolds numbers, flows tend to be dom ...
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
The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water.
Viscosity quantifies the int ...
and
thermal conductivity
The thermal conductivity of a material is a measure of its ability to conduct heat. It is commonly denoted by k, \lambda, or \kappa.
Heat transfer occurs at a lower rate in materials of low thermal conductivity than in materials of high thermal ...
.
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 The Lewis number (Le) is a dimensionless number defined as the ratio of thermal diffusivity to mass diffusivity. It is used to characterize fluid flows where there is simultaneous heat and mass transfer. The Lewis number puts the thickness of the ...
.
Experimental Values
Typical Values
For most gases over a wide range of temperature and pressure, 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 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
Hydrogen is the chemical element with the symbol H and atomic number 1. Hydrogen is the lightest element. At standard conditions hydrogen is a gas of diatomic molecules having the formula . It is colorless, odorless, tasteless, non-to ...
* 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
Water (chemical formula ) is an inorganic, transparent, tasteless, odorless, and nearly colorless chemical substance, which is the main constituent of Earth's hydrosphere and the fluids of all known living organisms (in which it acts as ...
(At 18
°C
The degree Celsius is the unit of temperature on the Celsius scale (originally known as the centigrade scale outside Sweden), one of two temperature scales used in the International System of Units (SI), the other being the Kelvin scale. The d ...
)
* 13.4 and 7.2 for
seawater
Seawater, or salt water, is water from a sea or ocean. On average, seawater in the world's oceans has a salinity of about 3.5% (35 g/L, 35 ppt, 600 mM). This means that every kilogram (roughly one liter by volume) of seawater has appro ...
(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 for
Earth
Earth is the third planet from the Sun and the only astronomical object known to harbor life. While large volumes of water can be found throughout the Solar System, only Earth sustains liquid surface water. About 71% of Earth's sur ...
's
mantle
A mantle is a piece of clothing, a type of cloak. Several other meanings are derived from that.
Mantle may refer to:
*Mantle (clothing), a cloak-like garment worn mainly by women as fashionable outerwear
**Mantle (vesture), an Eastern Orthodox ve ...
.
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.
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.
Physical Interpretation
Small values of the Prandtl number, , means the thermal diffusivity dominates. Whereas with large values, , the momentum diffusivity dominates the behavior.
For example, the listed value for liquid mercury indicates that the
heat conduction
Conduction is the process by which heat is transferred from the hotter end to the colder end of an object. The ability of the object to conduct heat is known as its ''thermal conductivity'', and is denoted .
Heat spontaneously flows along a te ...
is more significant compared to
convection
Convection is single or multiphase fluid flow that occurs spontaneously due to the combined effects of material property heterogeneity and body forces on a fluid, most commonly density and gravity (see buoyancy). When the cause of the c ...
, so thermal diffusivity is dominant.
However, for engine oil, convection is very effective in transferring
energy
In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of ...
from an area in comparison to pure conduction, so momentum diffusivity is dominant.
The Prandtl numbers of gases are about 1, which indicates that both
momentum
In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass ...
and
heat
In thermodynamics, heat is defined as the form of energy crossing the boundary of a thermodynamic system by virtue of a temperature difference across the boundary. A thermodynamic system does not ''contain'' heat. Nevertheless, the term is ...
dissipate through the fluid at about the same rate. Heat diffuses very quickly in liquid metals () and very slowly in oils () relative to momentum. Consequently
thermal boundary layer is much thicker for liquid metals and much thinner for oils relative to
velocity boundary layer.
In heat transfer problems, the Prandtl number controls the relative thickness of the momentum and thermal
boundary layers
In physics and fluid mechanics, a boundary layer is the thin layer of fluid in the immediate vicinity of a bounding surface formed by the fluid flowing along the surface. The fluid's interaction with the wall induces a no-slip boundary condi ...
. When is small, it means that the heat diffuses quickly compared to the velocity (momentum). This means that for liquid metals the thermal boundary layer is much thicker than the velocity boundary layer.
In laminar boundary layers, the ratio of the thermal to momentum boundary layer thickness over a flat plate is well approximated by
:
where
is the thermal boundary layer thickness and
is the momentum boundary layer thickness.
For incompressible flow over a flat plate, the two
Nusselt number correlations are asymptotically correct:
:
:
where
is the
Reynolds number
In fluid mechanics, the Reynolds number () is a dimensionless quantity that helps predict fluid flow patterns in different situations by measuring the ratio between inertial and viscous forces. At low Reynolds numbers, flows tend to be dom ...
. These two asymptotic solutions can be blended together using the concept of the
Norm (mathematics)
In mathematics, a norm is a function from a real or complex vector space to the non-negative real numbers that behaves in certain ways like the distance from the origin: it commutes with scaling, obeys a form of the triangle inequality, and ...
:
:
See also
*
Turbulent Prandtl number
*
Magnetic Prandtl number The Magnetic Prandtl number (Prm) is a dimensionless quantity occurring in magnetohydrodynamics which approximates the ratio of momentum diffusivity (viscosity) and magnetic diffusivity. It is defined as:
:\mathrm_\mathrm = \frac = \frac = \frac
...
References
General references
*
{{NonDimFluMech
Convection
Dimensionless numbers of fluid mechanics
Dimensionless numbers of thermodynamics
Fluid dynamics