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

\mathrm{St} = \frac{h}{G c_p} = \frac{h}{\rho u c_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_L}}{\mathrm{Re_L}\,\mathrm{Sc}}[4]

\mathrm{St}_m = \frac{h_m}{\rho u}[4]

where

  • St_m is the mass Stanton number;
  • Sh_L is the Sherwood number based on length;
  • Re_L is the Reynolds number based on length;
  • 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

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]

\Delta_2 = \int_0^\infty \frac{\rho u}{\rho_\infty u_\infty} \frac{T - T_\infty}{T_s - T_\infty} d y

Then the Stanton number is equivalent to

\mathrm{St} = \frac{d \Delta_2}{d x}

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]

\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}

See also

Strouhal number, an unrelated number that is also often denoted as \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". The Aeronautical Journal. 111 (1122): 473–493. doi:10.1017/S0001924000004735. ISSN 0001-9240. S2CID 113438383. Archived 2010-12-02 at the Wayback Machine.
  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.. 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.