# 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).[1][2]:476 It is used to characterize heat transfer in forced convection flows.

## Formula

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

where

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

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

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.

${\displaystyle \mathrm {St} _{m}={\frac {\mathrm {Sh} }{\mathrm {Re} \,\mathrm {Sc} }}}$

${\displaystyle \mathrm {St} _{m}={\frac {h_{m}}{\rho _{m}u}}}$

where

• ${\displaystyle St_{m}}$ is the mass Stanton number;
• ${\displaystyle Sh}$ is the Sherwood number;
• ${\displaystyle Re}$ is the Reynolds number;
• ${\displaystyle Sc}$ is the Schmidt number;
• ${\displaystyle h_{m}}$ is defined based on a concentration difference (kg s−1 m−2);
• ${\displaystyle u}$ is the velocity of the fluid
• ${\displaystyle \rho _{m}}$ 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:[4]

${\displaystyle \Delta _{2}=\int _{0}^{\infty }{\frac {\rho u}{\rho _{\infty }u_{\infty }}}{\frac {T-T_{\infty }}{T_{s}-T_{\infty }}}dy}$

Then the Stanton number is equivalent to

${\displaystyle \mathrm {St} ={\frac {d\Delta _{2}}{dx}}}$

for boundary layer flow over a flat plate with a constant surface temperature and properties.[5]

### 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[6]

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

where

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

## References

1. Hall, Carl W. (2018). Laws and Models: Science, Engineering, and Technology. CRC Press. pp. 424–. ISBN 978-1-4200-5054-7.
2. Ackroyd, J. A. D. (2016). "The Victoria University of Manchester's contributions to the development of aeronautics" (PDF). The Aeronautical Journal. 111 (1122): 473–493. doi:10.1017/S0001924000004735. ISSN 0001-9240. Archived from the original (PDF) on 2010-12-02.
3. Bird, R. Byron; Stewart, Warren E.; Lightfoot, Edwin N. (2006). Transport Phenomena. John Wiley & Sons. p. 428. ISBN 978-0-470-11539-8.
4. Crawford, Michael E. (September 2010). "Reynolds number". TEXSTAN. Institut für Thermodynamik der Luft- und Raumfahrt - Universität Stuttgart. Retrieved 26 August 2019.
5. Kays, William; Crawford, Michael; Weigand, Bernhard (2005). Convective Heat & Mass Transfer. McGraw-Hill. ISBN 978-0-07-299073-7.
6. Lienhard, John H. (2011). A Heat Transfer Textbook. Courier Corporation. p. 313. ISBN 978-0-486-47931-6.