# Simultaneous Conduction and Radiation Energy Transfer

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3 When T < Ts, Qc and Qr < 0, radiant energy flows out from lower radiator T to higher surroundings Ts. Qc = Qr; k(t - Ts) = 5.67 * (α εs Ts 4 - αs ε T 4 ), (4) Rearranging, kt αs ε T 4 = kts α εs Ts 4 ; LHS = RHS (5) Given Ts, we can find T, or vice versa. T = Ts, if and only if α εs = αs ε, (Qc = Qr = 0), which is rarely the case for dissimilar radiators. If αs ε > α εs, T < Ts by inspection. If αs ε < α εs, T > Ts. There will always be a solution T for any given Ts > 0. When both are black bodies, α = εs = αs = ε = 1. kt T 4 = kts Ts 4, T = Ts by inspection. If no other energy transfer is involved and Kirchhoff Law applies to both, α = ε, αs = εs kt εs ε T 4 = kt εs ε Ts 4, T = Ts by inspection. Poor Radiator Example Assume αs = 0.9, εs = 0.8, α = 0.6, ε = 0.5. αs ε = 0.45 < α εs = T * 0.9 * 0.5 T 4 = 100 Ts * 0.6 * 0.8 Ts T * 0.45 T 4 = 100 Ts * 0.48 Ts 4 Let Ts = 20C. We find T = about C > Ts RHS = 100 * * 0.48 * = * 0.48 * = = LHS = 100 * * 0.45 * = * 0.45 * = = , checks Qc = k(t - Ts) = 100 * ( ) = out Qr = 5.67 * (α εs Ts 4 - αs ε T 4 ) = 5.67 (0.6*0.8 * *0.5 * ) = 5.67 * (0.48 * * ) = 5.67 * ( ) = 5.67 * = in. Close to Qc, round-off. Note radiator T = > surroundings Ts = 20, but colder surroundings transfer radiant energy into warmer radiator, heating it because α εs > αs ε, 0.6*0.8 > 0.9*0.5. Qr = Qc =

7 A recent paper 2 derived rigorous equations for the coupled atmosphere and surface temperatures. Only system properties are needed, no empiricism. When one warming and three cooling mechanisms are included in the whole system, the net effect of CO2 on temperature is small and likely < 0. The remaining question to quantify the effect of CO2 changes on temperatures is: how much does CO2 affect the atmosphere s radiating properties: absorptivity and emissivity? Assuming 2 a 1% increase in the atmosphere s radiating properties, perhaps due to increased CO2, surface temperature change is C and atmosphere change is -0.39C. The net effect is slight cooling. Applying (3) we see Qr > 0 for transfer from surface up to and absorbed by atmosphere, even when Ts > Ta. That is because there is no conduction involved, as proved above. The presence of radiating CO2 increases atmosphere emissivity, a resistance to radiant energy transfer. CO2 is not an energy blocker or trapper; it is an absorber and transmitter. No radiant energy transfers from cold atmosphere with CO2 down to warm surface, warming the surface. A shiny white car has greater reflectivity than rough black car. So it has lower absorptivity and emissivity. Since white absorbs less radiant energy than black, it emits less, causing it to be cooler. Increase Ts case Surroundings warm up. Assume αs = 0.9, εs = 0.8, α = 0.6, ε = 0.5 as before. Let Ts = 30C. We find T = about C > Ts RHS = 100 * * 0.48 * = * 0.48 * = = LHS = 100 * * 0.45 * = * 0.45 * = = , checks Qc = k(t - Ts) = 100 * ( ) = out Qr = 5.67 * (α εs Ts 4 - αs ε T 4 ) = 5.67 (0.6*0.8 * *0.5 * = 5.67 * (0.48 * * ) = 5.67 * ( ) = 5.67 * = in. Close to Qc, round-off. Note radiator T > surroundings Ts, but colder surroundings transfer radiant energy into warmer radiator, heating it because α εs > αs ε, 0.6*0.8 > 0.9*0.5. T Ts increases from to or C when Ts increases from 20 to 30C. Put one hand on a mirror or metal faucet and the other on a fiberboard wall. The wall is warmer because it is a better absorber/emitter. Mirror and metal are better reflectors. Table of Solutions α ε αs εs α εs αs ε Ts T T - Ts

11 1. Hertzberg, M, The Night Time Radiative Transport Between the Earth s Surface, Its Atmosphere, and Free Space, Energy & Environment, V23, n5, 2012, p Latour, PR, Radiation Physics Laws Give the Effect of CO2 on Earth s Temperatures A Primer, Principia Scientific International, February 25,

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