Grashof Number Calculator
Find the Grashof number Gr = gβΔTL³/ν², the natural-convection analog of the Reynolds number.
🌡️ What is the Grashof Number Calculator?
This Grashof number calculator finds Gr = gβΔTL³/ν², the dimensionless number that governs natural (free) convection, the same way the Reynolds number governs forced convection. Enter the fluid's thermal expansion coefficient, the temperature difference between a surface and the surrounding fluid, a characteristic length, and kinematic viscosity, and it returns Gr along with the flow regime.
The Grashof number is the natural-convection analog of the Reynolds number: instead of comparing inertial forces to viscous forces (like Reynolds), it compares buoyancy forces to viscous forces. Buoyancy arises because a heated (or cooled) fluid near a surface has a different density than the surrounding fluid, and that density difference drives fluid motion without any external pump or fan. Gr determines whether that buoyancy-driven flow stays smooth and laminar or transitions toward turbulence, with roughly 10⁸ to 10⁹ marking the transition for a vertical plate.
Gr rarely stands alone in practice: multiplying it by the Prandtl number gives the Rayleigh number, Ra = Gr·Pr, which is the parameter that actually determines which natural-convection Nusselt number correlation to use. Use the Prandtl Number Calculator to find Pr for your fluid, then combine it with the Gr computed here to get Ra for the next step of a natural-convection heat transfer analysis.
This calculator is useful for engineers and students analyzing natural convection off heated electronics, building walls, radiators, pipes, and any surface where buoyancy, not a fan or pump, drives the airflow or fluid motion.
📐 Formula
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 - Air natural convection off a vertical heated plate
Example 2 - Water natural convection, heated vertical wall
Example 3 - Small electronics component, air cooling
❓ Frequently Asked Questions
🔗 Related Calculators
What is the Grashof number?
The Grashof number, Gr, is a dimensionless quantity that compares buoyancy forces to viscous forces in a fluid experiencing natural (free) convection. It is the natural-convection counterpart of the Reynolds number, which instead compares inertial to viscous forces in forced flow.
What is the formula for the Grashof number?
Gr = gβΔTL³/ν², where g is gravitational acceleration (9.81 m/s²), β is the fluid's thermal expansion coefficient (1/K), ΔT is the temperature difference between the surface and the surrounding fluid (K), L is a characteristic length (m), and ν is kinematic viscosity (m²/s).
What does the Grashof number tell you about flow regime?
A low Grashof number indicates laminar natural-convection flow dominated by viscous forces, while a Grashof number above roughly 10⁸ to 10⁹ for a vertical plate signals transition toward turbulent natural convection dominated by buoyancy-driven instabilities.
How is the Grashof number related to the Rayleigh number?
The Rayleigh number is defined as Ra = Gr·Pr, the product of the Grashof number and the Prandtl number. Ra, not Gr alone, is the parameter that determines which natural-convection heat transfer correlation (for the Nusselt number) applies to a given problem.
What is the thermal expansion coefficient β and how do I find it?
β measures how much a fluid's volume changes with temperature at constant pressure, in units of 1/K. For an ideal gas, β simplifies to 1/T, where T is the absolute ambient temperature in Kelvin. For liquids like water, β must come from a property table since it does not follow the ideal-gas shortcut.
Why does the Grashof number use length cubed instead of length?
Buoyancy force scales with the volume of fluid set in motion (proportional to L³), while viscous drag scales differently, so the ratio that defines Gr naturally involves L³. This is why doubling the characteristic length increases Gr by a factor of 8, not 2.
Is the Grashof number the same as the Reynolds number?
No. The Reynolds number compares inertial forces to viscous forces in forced convection (flow driven by an external pump or fan), while the Grashof number compares buoyancy forces to viscous forces in natural convection (flow driven purely by density differences from heating or cooling).
What is a typical Grashof number for air natural convection off a heated wall?
For air natural convection off a vertical heated plate about 0.5 m tall with a 30 K temperature difference, Gr comes out to roughly 5.5×10⁸, right around the laminar-to-turbulent transition, a common teaching example in heat transfer courses.
Does the Grashof number apply to forced convection?
No, the Grashof number is specific to natural (free) convection, where fluid motion is caused entirely by buoyancy from density differences. In mixed convection, where both buoyancy and an external forced flow matter, engineers compare Gr/Re² to determine which effect dominates.
Why is a higher Grashof number associated with more effective heat transfer?
A higher Grashof number means buoyancy forces increasingly dominate over viscous damping, driving stronger fluid circulation near the heated or cooled surface. This stronger circulation continuously replaces warmed (or cooled) fluid near the surface with fresh fluid, increasing the natural-convection heat transfer rate.