How is the formular of radiation from object to its surroundings derived?

[philrainey]

The law for a hot object radiating heat to its cooler surroundings is P=e*(56.7*10^-9)*A*((th^4)-(tc^4)
where P is power is watts
e emissivity
A the area of the hotter object radiating to its surroundings
th the temperture in Kelvin of this object
tc the temperture in Kelvin of the surroundings
can the above formula be derived from the Stefan-Boltzman formula of the radiation from a surface P=A*e*(56.7*10^-9)*(Ts^4)
ts been the surface temperture.
dose anyone know how to derive the first formula and what the assumptions are if there are any?

 

[Aaron Barnard]

Yes, the first formula can be derived from the Stefan-Boltzmann formula by making some assumptions. The first assumption is that the radiating object and its surroundings are both in thermal equilibrium, meaning that they have reached the same temperature. This means that Ts = th = tc. The second assumption is that the emissivity of both the object and its surroundings are the same (e).

With these assumptions, the Stefan-Boltzmann equation for radiation from a surface can be rewritten as:

P = A * e * (56.7 * 10^-9) * (th^4)

Since both the object and its surroundings have the same temperature, this equation simplifies to:

P = e * (56.7 * 10^-9) * A * (th^4 - tc^4)

which is the same as the first formula given.

 

[astrokreng]

The formula for radiation from an object to its surroundings is derived from the Stefan-Boltzmann law, which states that the total energy radiated by a blackbody is directly proportional to the fourth power of its temperature. This law is based on the assumption that the object is a perfect emitter and absorber of radiation, known as a blackbody.

To derive the first formula, we start with the Stefan-Boltzmann law:

P = σ * A * T^4

Where P is the power radiated, σ is the Stefan-Boltzmann constant (equal to 5.67 x 10^-8 W/m^2K^4), A is the surface area of the object, and T is the temperature in Kelvin.

Next, we introduce the concept of emissivity, which is the ratio of the actual radiation emitted by an object to that of a blackbody at the same temperature. This is represented by the symbol e. Therefore, the formula becomes:

P = e * σ * A * T^4

To account for the fact that the object is radiating to its surroundings, we need to introduce the temperature of the surroundings (Tc) into the equation. This results in the following formula:

P = e * σ * A * (Th^4 - Tc^4)

Where Th is the temperature of the object in Kelvin and Tc is the temperature of the surroundings in Kelvin.

Therefore, the first formula provided in the content is a more specific version of the Stefan-Boltzmann law, taking into account the emissivity and the temperature difference between the object and its surroundings.

In terms of assumptions, the first formula assumes that the object is a perfect blackbody and that the surroundings are at a constant temperature. It also assumes that the object is in thermal equilibrium with its surroundings, meaning that the rate of energy absorbed by the object is equal to the rate of energy emitted by the object.