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 Edward Stanton (1865–1931).^[1] It is used to characterize heat transfer in forced convection flows.

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

where

h = convection heat transfer coefficient
ρ = density of the fluid
c_p = specific heat of the fluid
u = speed of the fluid

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

Nu is the Nusselt number;
Re is the Reynolds number;
Pr is the Prandtl number.^[2]

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);
$\rho _{m}$ 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 ^[3]

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

Then the Stanton number is equivalent to ^[4]

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

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 ^[5]

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

where

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

References

↑ The Victoria University of Manchester’s contributions to the development of aeronautics
↑ Bird, Stewart, Lightfoot (2007). Transport Phenomena. New York: John Wiley & Sons. p. 428. ISBN 978-0-470-11539-8.
↑ Reynolds Number
↑ Kays, Crawford, Weigand (2005). Convective Heat and Mass Transfer. McGraw-Hill.
↑ Lienhard, Lienhard (2012). A Heat Transfer Texbook. Phlogiston Press.

Dimensionless numbers in fluid mechanics

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