Engineering
Carnot Efficiency & Stefan-Boltzmann Thermal Radiation formulas (Kelvin)
Calculate a two-reservoir Carnot bound and gray-surface radiation to large isothermal surroundings using absolute temperatures.
Never plug Celsius directly into Carnot or T^4 radiation—convert to Kelvin (+273.15) first!
Both foundational laws of thermodynamics require absolute temperature T(K) = T(C) + 273.15: Sadi Carnot's upper bound on any heat engine operating between hot reservoir T_H and cold sink T_C is eta_max = 1 - (T_C + 273.15) / (T_H + 273.15).
Similarly, Stefan-Boltzmann thermal radiation scales with the FOURTH POWER of absolute Kelvin temperature: emitted radiative power P = emissivity * sigma * Area * (T_H + 273.15)^4, and net radiative heat loss to ambient walls at T_C is P_net = emissivity * sigma * Area * ((T_H + 273.15)^4 - (T_C + 273.15)^4), where sigma = 5.670374419E-8 W/(m^2*K^4)!
The Carnot bound applies to a heat engine exchanging heat with two constant-temperature reservoirs; actual efficiency is lower for irreversible operation. The net radiation expression assumes an opaque diffuse-gray surface enclosed by much larger isothermal surroundings, with view factor effectively one. It is not the general exchange formula for two arbitrary finite surfaces, and it omits convection and conduction.
| Thermodynamic Input | Cell | Value | Kelvin & Output |
|---|---|---|---|
| Hot Surface / Reservoir (T_H) | B2 | 127 C | 400.15 K |
| Cold Ambient Sink (T_C) | B3 | 27 C | 300.15 K -> Carnot = 0.2499 (24.99%) |
| Surface Emissivity (eps) | B4 | 0.90 | Oxidized steel / painted casing |
| Radiating Area (m^2) | B5 | 2.0 m^2 | Emitted = 2,616.83 W | Net = 1,788.43 W |
Carnot maximum thermodynamic efficiency 1 - (T_C + 273.15) / (T_H + 273.15)
Between 127 C (400.15 K) and 27 C (300.15 K), no heat engine can exceed 100 / 400.15 = 24.99% efficiency.
=IF(OR(COUNT(B2:B3)<>2,B2<=-273.15,B3<=-273.15,B2<B3),"Invalid inputs",ROUND(1 - (B3 + 273.15) / (B2 + 273.15), 4))Stefan-Boltzmann emitted & net radiative heat transfer (Watts)
Use scientific notation 5.670374419E-8 for the CODATA Stefan-Boltzmann constant sigma.
Use temperatures above absolute zero, emissivity between zero and one, and nonnegative area. Net power is signed: a cooler surface gains heat. For the Carnot heat-engine calculation require hot temperature at least as high as cold temperature. Emissivity 0.9 is an assumed value, not a universal material constant.
=IF(OR(COUNT(B2:B5)<>4,B2<=-273.15,B3<=-273.15,B4<0,B4>1,B5<0),"Invalid inputs",ROUND(B4 * 5.670374419E-8 * B5 * (((B2 + 273.15) ^ 4) - ((B3 + 273.15) ^ 4)), 2))Try the example
Use the original values or change them to test your understanding.