Stanton number

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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.

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

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).

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

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]

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

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

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