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Engineering

Grashof Number (Gr) & Rayleigh Number (Ra) Natural Convection

Calculate ideal-gas Grashof and Rayleigh magnitudes with SI inputs, then compare Ra with an illustrative vertical-plate reference without treating it as a universal transition rule.

Use properties and geometry appropriate to the convection model

For the ideal-gas example, beta=1/Tfilm with absolute film temperature in kelvin. Gr magnitude is g beta |Ts-Tambient| L³/nu² and Ra=Gr Pr. L is the vertical plate height in metres, nu is kinematic viscosity in m²/s, and Pr is dimensionless. Enter mutually consistent fluid properties at the chosen reference temperature.

The spreadsheet uses the magnitude of the temperature difference. A colder plate produces the same numerical magnitude for the same properties, but buoyancy direction changes. Tfilm is the arithmetic mean in this example; Tfilm±deltaT/2 must remain positive kelvin. The ideal-gas beta formula is not a general liquid expansion coefficient.

Ra around 1e9 is a common reference in vertical-plate correlations, not an abrupt universal laminar/turbulent boundary. Orientation, geometry and correlation validity matter. The label only reports which side of that reference the computed number lies on; it does not predict plume structure or calculate a heat-transfer coefficient.

Doubling height multiplies Gr and Ra by eight only when all other inputs remain fixed. The model does not update viscosity or Pr automatically when temperature changes.

Unrounded Grashof magnitude in D2

The sign of deltaT is removed for this magnitude calculation. Zero deltaT gives zero thermal driving.

=IF(COUNT(B2:B6)<>5,"Enter numeric convection inputs",IF(OR(B2<=0,B3<=0,B5<=0,B6<=0,ABS(B4)>=2*B3),"Require positive properties and Kelvin endpoints",B2*(1/B3)*ABS(B4)*B5^3/B6^2))

Unrounded Rayleigh magnitude in D3

Use the unrounded Gr value so the threshold comparison does not depend on a rounded intermediate.

=IF(NOT(ISNUMBER(D2)),D2,IF(COUNT(B7)<>1,"Enter numeric Prandtl number",IF(B7<=0,"Prandtl number must be positive",D2*B7)))

Vertical-plate reference comparison in D4

This is a numerical comparison with a reference, not a universal flow classification.

=IF(NOT(ISNUMBER(D3)),D3,IF(D3=0,"No thermal driving",IF(D3<1000000000,"Below reference Ra=1e9","At or above reference Ra=1e9")))

Try the example

Use the original values or change them to test your understanding.

Further reading