# Prandtl Number

## The Prandtl Number is a dimensionless number approximating the ratio of momentum diffusivity to thermal diffusivity

The Prandtl Number is a dimensionless number approximating the ratio of *momentum diffusivity (kinematic viscosity)* to *thermal diffusivity* - and can be expressed as

Pr = v / α (1)

where

Pr = Prandtl's number

v = momentum diffusivity (m^{2}/s)

α = thermal diffusivity (m^{2}/s)

The Prandtl number can alternatively be expressed as

Pr = μ c_{p}/ k (2)

where

μ = absolute or dynamic viscosity (kg/m s,lb)_{m}/(ft h)

c_{p}= specific heat (J/kg K, Btu/(lb_{m}^{o}F))

k = thermal conductivity (W/m K, Btu/(h ft^{2}^{o}F/ft))

The Prandtl Number is often used in heat transfer and free and forced convection calculations. It depends on the fluid properties.

- gases -
*Pr ranges 0.7 - 1.0* - dry air
- water -
*Pr ranges 1 - 10* - liquid metals -
*Pr ranges 0.001 - 0.03* - oils -
*Pr ranges 50 - 2000*

### Example - Calculation of a Prandtl Number

*μ = 15 (cP) *

*c _{p} = 0.50 Btu/(lb_{m} ^{o}F)*

*k = 0.06 (Btu/(h ft ^{2} ^{o}F/ft)*

Dynamic viscosity can be converted from *cP* to * lb_{m}/(ft h)* as

*μ = 15 (cP) ( 2.4191 (lb_{m}/(ft hr)) / (1 cP))*

* = 36.3 lb/(ft hr)*

The Prandtl Number can be calculated as

*Pr = 36.3 (lb/(ft hr)) 0.50 Btu/(lb_{m} ^{o}F) / 0.06 (Btu/(h ft^{2} ^{o}F/ft)*

* = 302*