Physics: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
[math]\displaystyle{ St = \frac{h}{G c_p} = \frac{h}{\rho u c_p} }[/math]
where
- h = convection heat transfer coefficient
- ρ = density of the fluid
- cp = 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:
- [math]\displaystyle{ \mathrm{St} = \frac{\mathrm{Nu}}{\mathrm{Re}\,\mathrm{Pr}} }[/math]
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.
[math]\displaystyle{ \mathrm{St}_m = \frac{\mathrm{Sh_L}}{\mathrm{Re_L}\,\mathrm{Sc}} }[/math][4]
[math]\displaystyle{ \mathrm{St}_m = \frac{h_m}{\rho u} }[/math][4]
where
- [math]\displaystyle{ St_m }[/math] is the mass Stanton number;
- [math]\displaystyle{ Sh_L }[/math] is the Sherwood number based on length;
- [math]\displaystyle{ Re_L }[/math] is the Reynolds number based on length;
- [math]\displaystyle{ Sc }[/math] is the Schmidt number;
- [math]\displaystyle{ h_m }[/math] is defined based on a concentration difference (kg s−1 m−2);
- [math]\displaystyle{ u }[/math] 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]
[math]\displaystyle{ \Delta_2 = \int_0^\infty \frac{\rho u}{\rho_\infty u_\infty} \frac{T - T_\infty}{T_s - T_\infty} d y }[/math]
Then the Stanton number is equivalent to
[math]\displaystyle{ \mathrm{St} = \frac{d \Delta_2}{d x} }[/math]
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]
[math]\displaystyle{ \mathrm{St} = \frac{C_f / 2}{1 + 12.8 \left( \mathrm{Pr}^{0.68} - 1 \right) \sqrt{C_f / 2}} }[/math]
where
[math]\displaystyle{ C_f = \frac{0.455}{\left[ \mathrm{ln} \left( 0.06 \mathrm{Re}_x \right) \right]^2} }[/math]
See also
Strouhal number, an unrelated number that is also often denoted as [math]\displaystyle{ \mathrm{St} }[/math].
References
- ↑ Hall, Carl W. (2018). Laws and Models: Science, Engineering, and Technology. CRC Press. pp. 424–. ISBN 978-1-4200-5054-7. https://books.google.com/books?id=EEhpsf6L09gC&pg=PA424.
- ↑ 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. http://www.raes.org.uk/pdfs/3164COLOUR.pdf.
- ↑ Bird, R. Byron; Stewart, Warren E.; Lightfoot, Edwin N. (2006). Transport Phenomena. John Wiley & Sons. p. 428. ISBN 978-0-470-11539-8. https://books.google.com/books?id=L5FnNlIaGfcC&pg=PA428.
- ↑ 4.0 4.1 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.
- ↑ Crawford, Michael E. (September 2010). "Reynolds number". TEXSTAN. Institut für Thermodynamik der Luft- und Raumfahrt - Universität Stuttgart. http://www.texstan.com/ef1.php.
- ↑ Kays, William; Crawford, Michael; Weigand, Bernhard (2005). Convective Heat & Mass Transfer. McGraw-Hill. ISBN 978-0-07-299073-7. https://books.google.com/books?id=hiEmjxP6hQkC.
- ↑ Lienhard, John H. (2011). A Heat Transfer Textbook. Courier Corporation. p. 313. ISBN 978-0-486-47931-6. https://books.google.com/books?id=P8iV6IjNtI8C&pg=PA313.
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