Understanding the Stefan-Boltzmann Law (when the surroundings are hotter)

[JC2000]

TL;DR

1.My book tells me that given $T_{surroundings}$, and $T$ of the object radiating heat, the law is expressed as $H = \sigma A (T^4 - T^4_{surroundings})$.

2. Relating Newton's Law of Cooling, Conduction, and Stefan-Boltzmann Law

3. Is emissivity the same as Stefan's constant or is it $e * \sigma$ where $e$ varies, depending on the material?

1.If so what would the law mean if $T_{surroundings}>T$?

2. Stefan-Boltzmann Law is formulated as $H = A\sigma T^4$ where $H$ is the energy emitted per unit time, $A$ is the area of the object, $T$ is the absolute temperature of the object and (3.) I am unclear about whether $\sigma$ represents emissivity or $e*\sigma$ represents Stefan's constant.

My book also defines Conduction (as the time rate of heat flow for a given temperature difference), as $H = kA \frac {T_c - T_d}{L}$ where $H$ is the rate of flow of heat (heat current), $A$ is the area of cross-section and $L$ is the length between the two points being considered, $T_c - T_d$ the temperature difference between the points.

Newton's Law of Cooling is stated as a special case of Stefan-Boltzmann Law where the temperature difference is very small and is formulated as $\frac {dQ}{dt} = k(T_2 - T_1)$.

I feel that the three must be somehow interrelated, am I correct in assuming this? If so, how?

3. Lastly is emissivity the same as Stefan's constant or is it $e * \sigma$ where $e$ varies, depending on the material?

Thank you!

 

[256bits]

Hi JC
1. the object has a net absorption of radiation
2. e represents emissivity, σ represents the Stefan - Boltzmann constant
3. ε = 1 for a black body. For all real materials, it varies from 0 to 1, and can be frequency dependent.

Newton's Law of cooling - if the temperature difference is large, then radiation effects should be taken into account

 

[JC2000]

1. So $H$ represents the rate of absorption? (If so, shouldn't heat absorption have a positive sign and hence rate of heat absorption should also have a positive sign?)
2. Thanks that makes it clear!
3. So, it is safe to say that Stefan-Boltzmann's law could be used in place of the other two formulas while the other two are special cases (for small T differences we have Newton's Law of Cooling and the formula of conduction is when heat transfer is occurring within a substance). Thus Stefan-Boltzmann's law could be used to find the rate of heat transfer in all cases?

 

[256bits]

1. So $H$ represents the rate of absorption? (If so, shouldn't heat absorption have a positive sign and hence rate of heat absorption should also have a positive sign?)
2. Thanks that makes it clear!
3. So, it is safe to say that Stefan-Boltzmann's law could be used in place of the other two formulas while the other two are special cases (for small T differences we have Newton's Law of Cooling and the formula of conduction is when heat transfer is occurring within a substance). Thus Stefan-Boltzmann's law could be used to find the rate of heat transfer in all cases?

1. In which direction you assign heat flow, in or out, would be case dependent.
Example - the sun outputs X amount of radiation, so one would consider that a positive value for the sun.
the Earth receives Y amount of heat flux, so one would consider that as a positive value for the earth.

3. probably not. Conduction and convection are heat transfers by contact, not by radiation.