Rayleigh Number Calculator

At a glance:The Rayleigh number (symbol Ra) is the most important dimensionless number in natural-convection heat transfer, defined as the product of the Grashof and Prandtl numbers: Ra=Gr·Pr=(g·β·ΔT·L³/ν²)·(ν/α)=g·β·ΔT·L³/(ν·α), where Ra is the Rayleigh number (dimensionless), Gr the Grashof number (buoyancy vs viscous forces), Pr the Prandtl number (momentum vs thermal diffusion). Term by term: the Grashof number Gr describes how strongly buoyancy (driving natural convection) compares to viscous forces; the Prandtl number Pr describes the fluid's own momentum/thermal diffusion. Their product Ra contains both 'buoyant driving' and 'thermal/momentum diffusion', so it more directly reflects 'the strength of natural-convection heat transfer'. Physical meaning: when Ra is small, buoyancy cannot overcome viscosity and thermal diffusion, and heat passes mainly by conduction (the fluid barely moves); once Ra exceeds a critical value (horizontal fluid layer Ra_c≈1708), convection 'starts' and heat begins to move by fluid motion; larger Ra means stronger convection, and eventually turbulence. Example: 300 K air natural convection Gr≈2.65×10⁶, Pr≈0.7 → Ra=2.65×10⁶×0.7≈1.855×10⁶, in the laminar natural-convection range. Natural-convection correlations almost all use Ra as the independent variable, e.g. vertical plate Nu=0.59·Ra^(1/4) (laminar), Nu=0.1·Ra^(1/3) (turbulent). Uses: (1) judge whether natural convection starts (compare with the critical Rayleigh number); (2) as the independent variable in natural-convection correlations to get Nusselt number and heat-transfer coefficient; (3) distinguish laminar/turbulent natural convection; (4) analyse Benard convection, mantle convection, atmospheric convection. Notes: (1) take Gr and Pr at the same fluid and temperature; (2) different geometries have different critical Ra and correlations; (3) this tool multiplies Gr by Pr directly; (4) all quantities are dimensionless.