Library / Dimensional analysis and similarity / DIM-16
Dimensional analysis and similarity
Stanton number
Heat actually transferred to the fluid compared with the heat the stream can carry.
- Ref
- DIM-16
- Rev
- v1
- Field
- Dimensional analysis and similarity
- Result
- St
Sth = h/(ρcpU) compares the heat transferred to the fluid with the heat the stream can carry, ρcpU per unit area for each kelvin. It equals Nu/(Re Pr), so it can be found from a Nusselt correlation. For Prandtl numbers near 1 it is about half the skin-friction coefficient (the Reynolds analogy). The mass-transfer analogue is Stm = km/U = Sh/(Re Sc).
Heat actually transferred to the fluid compared with the heat the stream can carry. It is the Nusselt number divided by the product of the Reynolds and Prandtl numbers.
Calculator
- Stream heat-capacity rate per unit area, ρcpU12,000 W/(m² K)
- Skin-friction coefficient from the Reynolds analogy, 2St (Pr ≈ 1)0.01
- Stanton number with the speed doubled and h unchanged0.0025
- Heat flux per kelvin at the surface60 W/(m² K)
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- St = 0.005 means the surface passes a fraction 0.005 of the stream’s heat-carrying capacity per kelvin: 0.5 % of the maximum possible.
- Combined with the Reynolds analogy, St ≈ Cf/2 when Pr is near 1, so a skin-friction measurement gives a first estimate of the heat-transfer coefficient.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| h Convective heat-transfer coefficient | 1 to 10,000 W/(m² K) | 8.333e-5 | 0.8333 | 0.8333 | +10% | +10 % |
| U Free-stream or bulk speed | 0.1 to 400 m/s | 1.25e-4 | 0.5 | 0.4999 | +10% | −9.09 % |
| ρ Fluid density | 0.1 to 2,000 kg/m³ | 3e-6 | 0.06 | 0.06 | +10% | −9.09 % |
| cp Specific heat capacity at constant pressure | 500 to 5,000 J/(kg K) | 0.001 | 0.01 | 0.009 | +10% | −9.09 % |
Sweep of Free-stream or bulk speed
| Free-stream or bulk speed (m/s) | Stanton number |
|---|---|
| 0.1 | 0.5 |
| 0.282 | 0.1773 |
| 0.7953 | 0.06287 |
| 2.243 | 0.02229 |
| 6.325 | 0.007906 |
| 17.84 | 0.002803 |
| 50.3 | 9.941e-4 |
| 141.8 | 3.525e-4 |
| 400 | 1.25e-4 |
Sweep one variable
Variables and how each one contributes
Stanton number
What the equation gives you. Everything else feeds it.
Convective heat-transfer coefficient
The convective heat-transfer coefficient, in W/(m² K). Raises the Stanton number in direct proportion. Raising it by 10 % (to 66) moves the result by +10 %, from 0.005 to 0.0055.
Fluid density
The fluid density, in kg/m³. Lowers the Stanton number in inverse proportion. Raising it by 10 % (to 1.32) moves the result by −9.09 %, from 0.005 to 0.004545.
Specific heat capacity at constant pressure
The specific heat capacity at constant pressure, in J/(kg K) (air about 1005, water about 4180). Lowers the Stanton number in inverse proportion. Raising it by 10 % (to 1,100) moves the result by −9.09 %, from 0.005 to 0.004545.
Free-stream or bulk speed
The free-stream or bulk speed, in m/s. Lowers the Stanton number in inverse proportion: the stream carries more heat at higher speed. Raising it by 10 % (to 11) moves the result by −9.09 %, from 0.005 to 0.004545.
Worked example with the values above
- Stream capacity: ρcpU = 1.2 × 1,000 × 10 = 12,000 W/(m² K).
- Divide the heat-transfer coefficient by it: 60 / 12,000.
- St = 0.005.
Assumptions and limits
- h is the convective coefficient defined with the same reference temperature difference as in the correlation used.
- Properties are evaluated at a consistent reference temperature, such as the film temperature.
Common mistakes
- Using the specific heat at constant volume instead of cp.
- Mixing U from different positions: free-stream or bulk.
- Using the mass-transfer form Stm = km/U with a thermal h.
Statistics and uncertainty
The result is a product of powers, so to first order its relative error is the root sum of squares of each input’s relative error multiplied by its exponent (h × 1, ρ × −1, cp × −1, U × −1). The coefficient h usually carries the largest uncertainty, typically 10 to 25 % from a correlation, so St inherits it.
Where it is used
- Expressing a convective heat-transfer coefficient as a size-independent number for correlations and for scaling.
- Relating heat transfer to skin friction in boundary layers through the Reynolds analogy.
Maths you need
- Nusselt number: dimensionless heat-transfer coefficient
- Specific heat capacity: energy to heat 1 kg by 1 K