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

• h = convection heat transfer coefficient
• G = mass flux of the fluid
• ρ = density of the fluid
• c_p = specific heat of the fluid
• u = velocity of the fluid

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

• Nu is the Nusselt number;
• Re is the Reynolds number;
• Pr is the Prandtl number.^[3]

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.

${\displaystyle \mathrm {St} _{m}={\frac {\mathrm {Sh_{L}} }{\mathrm {Re_{L}} \,\mathrm {Sc} }}}$^[4]

${\displaystyle \mathrm {St} _{m}={\frac {h_{m}}{\rho u}}}$^[4]

where

• ${\displaystyle St_{m}}$ is the mass Stanton number;
• ${\displaystyle Sh_{L}}$ is the Sherwood number based on length;
• ${\displaystyle Re_{L}}$ is the Reynolds number based on length;
• ${\displaystyle Sc}$ 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

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:^[5]

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

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.^[6]

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^[7]

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

where

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

See also

Strouhal number, an unrelated number that is also often denoted as ${\displaystyle \mathrm {St} }$.

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. S2CID 113438383. 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. Fundamentals of heat and mass transfer. Bergman, T. L., Incropera, Frank P. (7th ed.). Hoboken, NJ: Wiley. 2011. ISBN 978-0-470-50197-9. OCLC 713621645.
5. Crawford, Michael E. (September 2010). "Reynolds number". TEXSTAN. Institut für Thermodynamik der Luft- und Raumfahrt - Universität Stuttgart. Retrieved 2019-08-26.
6. Kays, William; Crawford, Michael; Weigand, Bernhard (2005). Convective Heat & Mass Transfer. McGraw-Hill. ISBN 978-0-07-299073-7.
7. Lienhard, John H. (2011). A Heat Transfer Textbook. Courier Corporation. p. 313. ISBN 978-0-486-47931-6.