Lesson 6 - Radiation and Combined

The Stefan-Boltzmann law quantifies the energy radiated from or absorbed by a black body at a given temperature.

But what is a black body?

A black body emitter is the ‘ideal’ or ‘perfect’ emitter, it absorbs and emits all incoming radiation.

Nearly all matter in the universe is not ideal and cannot absorb/emit all incoming radiation. We call the non-ideal bodies ‘grey’ bodies.

The ratio of the emission from a non-ideal ’grey’ surface to the emission from a perfect ’black’ body is called the emissivity. Mathematically, emissivity is defined as:

Where

            W is the total emissive power of the specific substance in Wm^-2

            W_b is the total emissive power of a black body in Wm^-2

The surface of a black body emitter emits thermal radiation at a rate of roughly 448 Wm^-2 at 25°C.

The emissivity of surfaces ranges from 1, which is a perfect black body radiator, to 0 which is termed a perfect reflector.

Polished metal surfaces have relatively low emissivity and are consequently good reflectors. As a polished surface becomes degraded due to corrosion, its emissivity increases.

The emissivity of some common surfaces are given in the table below. 

Table: Emissivity of common materials

The radiating power is described by the Stefan-Boltzmann law. The energy emitted by a black body is proportional to the fourth power of the absolute temperature. 

Where

W_b is the total emissive power of a black body in Wm^-2

σ is the Stefan-Boltzmann constant = 5.672 × 10^-8 Wm^-2 K^-4

T is the absolute temperature in K

It is important to note that the Stefan-Boltzmann law requires the use of temperature in Kelvin.

If we want to know the total radiation for a unit area of a grey body, we need to take into account the area of the body and the emissivity of the surface of the body by using the following equation.

In a brewery setting, we may want to know the amount of heat absorbed or emitted by a piece of equipment. In that case, the radiation depends on the temperature difference between the two bodies.

Assume that an object is a grey body with emissivity ε and it is radiating heat to another body which is a black body absorber. We can then adapt the above equation to calculate the net rate of heat exchange between two bodies: 

Where

q is the heat transfer rate by radiation in Js^-1 (W)

A is the heat transfer area of radiating surface in m²

σ is the Stefan-Boltzmann constant = 5.672 × 10^-8 Wm^-2 K^-4 

ε is the emissivity of the grey body

T₁ and T₂ are the temperatures of the emitting surface and surrounding matter in K

A key point of the Stefan-Boltzmann law and its derivations is that the radiating power is dependent on temperature more than any other parameter. In all the equations, temperature is to the 4th power, meaning it has a large impact on the heat transfer rate by radiation.

We will discuss why this is important in a brewery in the next section.