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The '''Prandtl number''' <math>\mathrm{Pr}</math> is a [[dimensionless number]]; the ratio of momentum diffusivity ([[Viscosity#Kinematic viscosity|kinematic viscosity]]) to [[thermal diffusivity]]. It is named after the German physicist [[Ludwig Prandtl]].
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It is defined as:
:<math>\mathrm{Pr} = \frac{\nu}{\alpha} = \frac{\mbox{viscous diffusion rate}}{\mbox{thermal diffusion rate}} = \frac{c_p \mu}{k}</math>
 
where:
* <math>\nu</math> : [[viscosity#Kinematic viscosity|kinematic viscosity]], <math>\nu = \mu/\rho</math>, ([[SI]] units : m<sup>2</sup>/s)
* <math>\alpha</math> : [[thermal diffusivity]], <math>\alpha = k/(\rho c_p)</math>, (SI units : m<sup>2</sup>/s)
* <math>\mu</math> : [[dynamic viscosity]], (SI units : Pa s = N s/m<sup>2</sup>
* <math>k</math>: [[thermal conductivity]], (SI units : W/(m K) )
* <math>c_p</math> : [[specific heat]], (SI units : J/(kg K) )
* <math>\rho</math> : [[density]], (SI units : kg/m<sup>3</sup> ).  
 
Note that whereas the [[Reynolds number]] and [[Grashof number]] are subscripted with a length scale variable, the Prandtl number contains no such length scale in its definition and is dependent only 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]].  
 
Typical values for <math>\mathrm{Pr}</math> are:
* around 0.015 for [[mercury (element)|mercury]]
* around 0.16-0.7 for mixtures of [[noble gas]]es or noble gases with [[hydrogen]]
* around 0.7-0.8 for [[air]] and many other [[gas]]es,
* between 4 and 5 for [[Dichlorodifluoromethane|R-12 refrigerant]]
* around 7 for [[water]] (At 20 [[degrees Celsius]])
* 13.4 and 7.2 for [[seawater]] (At 0 [[degrees Celsius]] and 20 [[degrees Celsius]] respectively)
* between 100 and 40,000 for [[engine oil]]
* around 1{{e|25}} for [[Earth]]'s [[mantle (geology)|mantle]].
(<math>\mathrm{Pr}\ll 1</math> means thermal diffusivity dominates),(<math>\mathrm{Pr} \gg 1</math> means momentum diffusivity dominates)
 
For mercury, [[heat conduction]] is very effective compared to [[convection]]: thermal diffusivity is dominant.
For engine oil, convection is very effective in transferring [[energy]] from an area, compared to pure conduction: momentum diffusivity is dominant.
 
In heat transfer problems, the Prandtl number controls the relative thickness of the momentum and thermal [[boundary layers]]. When ''Pr'' is small, it means that the heat diffuses very quickly compared to the velocity (momentum). This means that for liquid metals the thickness of the thermal boundary layer is much bigger than the velocity boundary layer.
 
The mass transfer analog of the Prandtl number is the [[Schmidt number]].
 
==See also==
*[[Turbulent Prandtl number]]
*[[Magnetic Prandtl number]]
 
== References ==
*{{cite book |title=Viscous Fluid Flow |first=F. M. |last=White |location=New York |publisher=McGraw-Hill |edition=3rd. |year=2006 |isbn=0-07-240231-8 }}
 
{{NonDimFluMech}}
 
[[Category:Convection]]
[[Category:Dimensionless numbers of fluid mechanics]]
[[Category:Dimensionless numbers of thermodynamics]]
[[Category:Fluid dynamics]]

Latest revision as of 21:02, 2 January 2015

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