# Stanton number

The Stanton number, St, is a dimensionless number that measures the ratio of heat transferred into a fluid to the thermal capacity of fluid. The Stanton number is named after Thomas Stanton (engineer) (1865–1931).:476 It is used to characterize heat transfer in forced convection flows.

## Formula

$St={\frac {h}{Gc_{p}}}={\frac {h}{\rho uc_{p}}}$ where

It can also be represented in terms of the fluid's Nusselt, Reynolds, and Prandtl numbers:

$\mathrm {St} ={\frac {\mathrm {Nu} }{\mathrm {Re} \,\mathrm {Pr} }}$ where

The Stanton number arises in the consideration of the geometric similarity of the momentum boundary layer and the thermal boundary layer, where it can be used to express a relationship between the shear force at the wall (due to viscous drag) and the total heat transfer at the wall (due to thermal diffusivity).

## Mass transfer

Using the heat-mass transfer analogy, a mass transfer St equivalent can be found using the Sherwood number and Schmidt number in place of the Nusselt number and Prandtl number, respectively.

$\mathrm {St} _{m}={\frac {\mathrm {Sh} }{\mathrm {Re} \,\mathrm {Sc} }}$ $\mathrm {St} _{m}={\frac {h_{m}}{\rho _{m}u}}$ where

• $St_{m}$ is the mass Stanton number;
• $Sh$ is the Sherwood number;
• $Re$ is the Reynolds number;
• $Sc$ is the Schmidt number;
• $h_{m}$ is defined based on a concentration difference (kg s−1 m−2);
• $u$ is the velocity of the fluid
• $\rho _{m}$ is the component density of the species in flux.

## Boundary layer flow

The Stanton number is a useful measure of the rate of change of the thermal energy deficit (or excess) in the boundary layer due to heat transfer from a planar surface. If the enthalpy thickness is defined as:

$\Delta _{2}=\int _{0}^{\infty }{\frac {\rho u}{\rho _{\infty }u_{\infty }}}{\frac {T-T_{\infty }}{T_{s}-T_{\infty }}}dy$ Then the Stanton number is equivalent to

$\mathrm {St} ={\frac {d\Delta _{2}}{dx}}$ for boundary layer flow over a flat plate with a constant surface temperature and properties.

### Correlations using Reynolds-Colburn analogy

Using the Reynolds-Colburn analogy for turbulent flow with a thermal log and viscous sub layer model, the following correlation for turbulent heat transfer for is applicable

$\mathrm {St} ={\frac {C_{f}/2}{1+12.8\left(\mathrm {Pr} ^{0.68}-1\right){\sqrt {C_{f}/2}}}}$ where

$C_{f}={\frac {0.455}{\left[\mathrm {ln} \left(0.06\mathrm {Re} _{x}\right)\right]^{2}}}$ 