Grashof Number
Grashof number: Gr = (g * beta * delta-T * L^3) / nu^2.
Measure the ratio of buoyancy to viscous forces in natural convection heat transfer.
The Formula
The Grashof number is a dimensionless quantity that characterizes the importance of buoyancy-driven flow relative to viscous forces in a fluid. It is the governing parameter for natural convection — heat transfer driven by density differences caused by temperature variations, without any external pump or fan. Named after German engineer Franz Grashof, the number plays the same role in natural convection analysis that the Reynolds number plays in forced convection: it determines whether the buoyancy-driven flow is laminar or turbulent.
In the formula, g is gravitational acceleration (9.81 m/s²), β (beta) is the thermal expansion coefficient of the fluid, ΔT is the temperature difference between the hot surface and the surrounding fluid, L is the characteristic length of the surface (height for vertical plates, diameter for cylinders), and ν (nu) is the kinematic viscosity of the fluid. The numerator represents buoyancy forces, and the denominator represents viscous resistance to flow.
A high Grashof number means buoyancy forces dominate over viscous forces, and the natural convection flow is likely to be turbulent. A low Grashof number indicates viscous forces suppress the buoyancy-driven motion, resulting in laminar flow. For vertical plates, the transition from laminar to turbulent natural convection typically occurs around Gr ≈ 10⁹. The product of the Grashof number and the Prandtl number — called the Rayleigh number — is often used directly in heat transfer correlations: Ra = Gr × Pr.
The thermal expansion coefficient β for an ideal gas is simply 1/T (where T is the absolute temperature). For liquids, β must be found in tabulated fluid property data. Understanding the Grashof number is essential for designing cooling systems, building facades, solar collectors, and any application where heat moves through a fluid without mechanical assistance.
Variables
| Symbol | Meaning | Unit |
|---|---|---|
| Gr | Grashof number — ratio of buoyancy to viscous forces | dimensionless |
| g | Gravitational acceleration = 9.81 | m/s² |
| β | Thermal expansion coefficient of the fluid | 1/K |
| ΔT | Temperature difference between hot surface and bulk fluid | K or °C |
| L | Characteristic length (height, diameter, etc.) | m |
| ν | Kinematic viscosity of the fluid | m²/s |
Example 1
A vertical wall at 60°C is in contact with air at 20°C. The wall height is 0.5 m. Air properties at the film temperature (40°C): β = 1/313 K⁻¹ = 0.00319 K⁻¹, ν = 1.7 × 10⁻⁵ m²/s. Calculate the Grashof number.
ΔT = 60 − 20 = 40 K
Gr = (9.81 × 0.00319 × 40 × 0.5³) / (1.7 × 10⁻⁵)²
Gr = (9.81 × 0.00319 × 40 × 0.125) / (2.89 × 10⁻¹⁰)
Gr = 0.01563 / 2.89 × 10⁻¹⁰ ≈ 5.41 × 10⁷
Gr ≈ 5.4 × 10⁷ — laminar natural convection (below the turbulent transition of ~10⁹)
Example 2
A 2 m tall heated building facade has ΔT = 30 K with outside air. Same air properties as above. Is the convection laminar or turbulent?
Gr = (9.81 × 0.00319 × 30 × 2³) / (1.7 × 10⁻⁵)²
Gr = (9.81 × 0.00319 × 30 × 8) / 2.89 × 10⁻¹⁰
Gr ≈ 7.52 / 2.89 × 10⁻¹⁰ ≈ 2.6 × 10¹⁰
Gr ≈ 2.6 × 10¹⁰ — turbulent natural convection (well above the 10⁹ transition threshold)
When to Use It
Use the Grashof number when analyzing natural convection scenarios:
- Designing passive cooling systems for electronics, buildings, and solar collectors
- Calculating heat loss from heated surfaces in still air without fans or pumps
- Determining whether buoyancy-driven flow is laminar or turbulent before selecting heat transfer correlations
- Analyzing heat transfer from vertical plates, horizontal cylinders, and spheres in natural convection
- Comparing buoyancy effects against forced flow effects in mixed convection problems
- HVAC engineering and building envelope thermal performance calculations