Grashof Number Calculator

The Grashof number is what the Reynolds number would be if the driving force were buoyancy instead of an external pump or fan. It governs natural convection: heat moving through a fluid because warmer regions are less dense and rise on their own. Every wall radiator in a cold room, every hot plate cooling in still air, every solar collector convecting heat into a fluid loop is described by Gr.

The formula:

Gr = g · β · ΔT · L³ / ν²

Where g is gravity (9.81 m/s²), β is the thermal expansion coefficient of the fluid (1/K), ΔT is the temperature difference between surface and fluid (K), L is the characteristic length (height for a vertical plate, diameter for a horizontal cylinder, etc.), and ν is the kinematic viscosity (m²/s).

Interpreting it:

• Gr < 10⁴: Laminar natural convection or essentially no flow. Buoyancy is too weak to overcome viscous resistance. Conduction dominates.
• 10⁴ ≤ Gr < 10⁹: Laminar convection cells, stable thermal plumes. The most common regime for room-temperature applications: heated walls, radiators, electronics with passive cooling.
• Gr ≥ 10⁹: Turbulent natural convection. Chaotic plumes, well-mixed flow. Large outdoor surfaces, building facades on hot days.

The transition from laminar to turbulent for a vertical plate happens around Gr ≈ 10⁹. The transition value depends on geometry — horizontal cylinders, spheres, and inclined surfaces all have their own critical Grashof numbers.

Named after Franz Grashof:

Franz Grashof was a German mechanical engineering professor in Karlsruhe in the late 19th century. He was one of the founders of the Verein Deutscher Ingenieure (VDI), the influential German engineering society. The dimensionless group named after him was actually formalized later by others, but the convention stuck because Grashof had laid much of the theoretical groundwork.

Why two dimensionless numbers, not one?

For forced convection (pump or fan moves the fluid), Re tells you the flow regime and Pr the fluid character. For natural convection (buoyancy moves the fluid), Gr replaces Re. The Prandtl number still applies. Their product is the Rayleigh number:

Ra = Gr · Pr

Ra is the workhorse parameter in natural-convection correlations. A typical Nusselt correlation for a vertical plate looks like:

Nu = 0.59 · Ra^(1/4) for 10⁴ < Ra < 10⁹ (laminar) Nu = 0.10 · Ra^(1/3) for Ra > 10⁹ (turbulent)

The exponent shifts from 1/4 to 1/3 at the transition. Same form as forced-convection Dittus-Boelter, but Ra replaces Re·Pr.

Worked example, vertical wall in a room:

A 1.5 m tall vertical electric heater surface is at 60 °C. Room air is at 20 °C. For air, β ≈ 1/T_film where T_film is the mean of surface and bulk temperature. T_film = (60+20)/2 + 273.15 = 313.15 K, so β = 1/313.15 ≈ 3.19 × 10⁻³ /K. Kinematic viscosity of air at this temperature: ν ≈ 1.7 × 10⁻⁵ m²/s.

Gr = 9.81 × 3.19 × 10⁻³ × 40 × 1.5³ / (1.7 × 10⁻⁵)² = 9.81 × 3.19 × 10⁻³ × 40 × 3.375 / 2.89 × 10⁻¹⁰ = 4.222 / 2.89 × 10⁻¹⁰ = 1.46 × 10¹⁰

So Gr > 10⁹: turbulent natural convection along the heater. Using the turbulent Nu correlation with Pr_air = 0.71:

Ra = 1.46 × 10¹⁰ × 0.71 = 1.04 × 10¹⁰ Nu = 0.10 · (1.04 × 10¹⁰)^(1/3) = 0.10 · 2173 = 217

For air with k = 0.026 W/m·K and L = 1.5 m: h = Nu·k/L = 217 × 0.026 / 1.5 = 3.76 W/m²·K. This is a typical natural-convection h for a vertical wall in still air, matching textbook values of 3-5 W/m²·K. A small fan that boosts h to 30 W/m²·K (forced convection) delivers about 8× more heat per square meter for the same temperature difference, which is why forced cooling is so much more effective.

Worked example, hot pipe in water:

A horizontal pipe at 80 °C is submerged in 20 °C water. Pipe diameter D = 50 mm. For water at the film temperature, β ≈ 0.4 × 10⁻³ /K, ν ≈ 6 × 10⁻⁷ m²/s.

Gr = 9.81 × 0.4 × 10⁻³ × 60 × 0.05³ / (6 × 10⁻⁷)² = 9.81 × 0.4 × 10⁻³ × 60 × 1.25 × 10⁻⁴ / 3.6 × 10⁻¹³ = 2.94 × 10⁻⁵ / 3.6 × 10⁻¹³ = 8.17 × 10⁷

In the laminar regime. The same pipe in air at the same ΔT would have Gr ~ 10⁵ (much smaller, because air’s β/ν² ratio is far lower). This is one reason why water-cooled radiators move heat dozens of times faster than air-cooled ones at modest ΔT — the natural-convection-driven flow is much stronger in water.

β for ideal gases:

For an ideal gas, β = 1/T (with T in Kelvin), so β at 25 °C is about 1/298 ≈ 3.36 × 10⁻³ /K. For liquids, β is tabulated and roughly 0.1-1 × 10⁻³ /K for most fluids; it changes with temperature, so the film temperature (mean of surface and bulk) is the right value to use.

Variants and limits:

• Modified Grashof number Gr = Gr · q"·L / k* is used when heat flux is specified instead of surface temperature.
• For Pr → ∞ (very viscous fluids like glycerin), the laminar boundary layer is so thin that Ra^(1/4) underpredicts; modified correlations use Ra^(1/5) with Pr corrections.
• For Pr → 0 (liquid metals), the thermal boundary layer extends well beyond the velocity layer, and Gr-only correlations are reasonable, but more carefully one uses Boussinesq’s full equations.

Where Grashof shows up beyond convection:

• Geophysics: ocean and atmospheric circulation are large-scale natural-convection problems. The Earth’s atmosphere has Gr ≈ 10²⁰ at the global scale — wildly turbulent.
• Volcanology: magma chamber convection is characterized by Gr at the kilometer scale.
• Building HVAC: passive cooling design hinges on Gr; stack ventilation in tall atria is buoyancy-driven flow at Gr ~ 10⁸.
• Industrial process design: solar still, distillation column, crystallizer designs all use Gr to size natural-convection components.