Biot Number (Bi), Fourier Number (Fo) & Lumped-Capacitance Cooling

Use the correct resistance ratio and length

Bi = h*Lc/k compares internal conduction resistance with external convection resistance, not the reverse. For this lumped model, Lc is volume divided by the surface area exchanging heat. Use the same length in Bi and Fo.

Bi below about 0.1 is a common screening rule for small internal temperature gradients, not a universal guarantee of a particular error percentage. This workbook suppresses the lumped temperature when the unrounded Bi is at least 0.1. A spatial model must match the actual geometry and boundary conditions; a Heisler chart is not automatically appropriate.

Calculate Biot number

B2 is h in W/(m²·K), B3 is Lc in meters, and B4 is solid conductivity in W/(m·K). E2 holds the unrounded dimensionless result.

=IF(COUNT(B2:B8)<>7,"Complete numeric inputs",IF(OR(B2<0,B3<=0,B4<=0,B5<=0,B6<0,B7< -273.15,B8< -273.15),"Invalid physical input",B2*B3/B4))

Calculate Fourier number

B5 is thermal diffusivity in m²/s and B6 is elapsed time in seconds. Diffusivity must be consistent with conductivity, density and heat capacity: alpha=k/(rho*cp). Changing conductivity alone does not create a consistent model of a different material.

=IF(COUNT(B2:B8)<>7,"Complete numeric inputs",IF(OR(B2<0,B3<=0,B4<=0,B5<=0,B6<0,B7< -273.15,B8< -273.15),"Invalid physical input",B5*B6/B3^2))

Calculate the approximate uniform temperature

B7 and B8 are the initial and fixed ambient temperatures in °C. The solution assumes initially uniform temperature, constant properties and heat-transfer coefficient, convection over the specified surface, and no internal heat generation or phase change. Radiation is not separately modeled.

For the supplied values Bi=0.002 and Fo=400, giving approximately 97.40°C after 125 seconds. This is a solution of the lumped model, not an exact temperature field in a real component.

At time zero the result equals the initial temperature. Zero convection coefficient leaves temperature unchanged. The same expression describes heating toward a warmer environment; equal initial and ambient temperatures remain unchanged.

=IF(COUNT(B2:B8)<>7,"Complete numeric inputs",IF(OR(B2<0,B3<=0,B4<=0,B5<=0,B6<0,B7< -273.15,B8< -273.15),"Invalid physical input",IF(B2*B3/B4>=0.1,"Bi >= 0.1: review spatial model",B8+(B7-B8)*EXP(-(B2*B3/B4)*(B5*B6/B3^2)))))

Further reading

• BYU Heat Transfer — Lumped Capacitance
• COMSOL — Thermal Capacitor theory