Definition of Prandtl Number
Definition
The dimensionless quantity obtained by dividing the product of specific heat $C_{p}$ and dynamic viscosity $\mu$ by thermal conductivity $k$ is called the Prandtl number. $$ \mathrm{Pr} = \frac{C_p \mu}{k} $$
Explanation
An intuitive interpretation of the Prandtl number, as with other dimensionless quantities, is obtained by considering the factors proportional to its magnitude and separating numerator and denominator. $\mathrm{Pr}$ Being large means high specific heat, high viscosity, or low thermal conductivity. A relatively large $\mathrm{Pr} \gg 0$ implies, in the context of heat transfer, that the material acts well as an insulator.
- If the specific heat is high, it takes longer to heat the material itself.
- High viscosity means that convection is reduced.
- Low thermal conductivity means less energy transfer.
- Conversely, in contexts where heat transfer efficiency must be increased, a lower $\mathrm{Pr}$ is preferable.
On the other hand, if one considers density $\rho$ in both numerator and denominator, the thermal conductivity $k$ relates to the thermal diffusivity $\alpha$ by $k = \rho C_{P} \alpha$ and the viscosity $\mu$ relates to the kinematic viscosity $\nu$ by $\mu = \rho \nu$, so the Prandtl number can be written in the simpler form below. $$ \mathrm{Pr} = \frac{C_{P} \mu}{\rho C_{P} \alpha} = \frac{\mu / \rho}{\alpha} = \frac{\nu}{\alpha} $$
Relationship with other dimensionless numbers
The Schmidt number and the Prandtl number cancel the kinematic viscosity $\nu$ as follows, and by their ratio one can express the Lewis number. $$ \begin{align*} \mathrm{Sc} =& {\frac{ \nu }{ D }} \\ \mathrm{Pr} =& {\frac{ \nu }{ \alpha }} \\ \implies \mathrm{Le} =& = {\frac{ \nu }{ D }} \left( {\frac{ \nu }{ \alpha }} \right)^{-1} = \frac{ \mathrm{Sc} }{ \mathrm{Pr} } \end{align*} $$