In the field of heat transfer, there is a lot of notation pertaining to heat and how it is being transferred or received. Let's take a look at some of the different variables and what they stand for.
Some common variables that will appear very often in this guide are
Now that we have defined some common nomenclature, let's start learning about the 3 modes of heat transfer. First up is conduction.
Conduction is the random movement of constituent atoms, molecules, and/or electrons in a solid body or stagnant fluid and hence requires contact for heat transfer to occur. Conduction rate can be described by what's known as Fourier's Law.
[ 1 ]
$$ \bar{q}^" = -k \nabla T $$
$ \bar{q}^" $ is what's known as the conduction heat flux ($ \frac{W}{m^2} $), $ k $ is the thermal conductivity ($ \frac{W}{m•K} $), and $ \nabla T $ is the temperature gradient ($ \frac{K}{m} $). One of the nicest examples to work with when considering conduction is one-dimensional, steady conduction with constant thermal conductivity.
In this case, we can define the conduction heat flux as
[ 2 ]
$$ q"_x = -k \frac{dT}{dx} = -k \frac{T_2 - T_1}{L} $$
You can picture this best using Fig. 1 below.
Fig. 1 - Example of 1-D, steady conduction across a plane wall.
In Figure 1, there is a plane wall with length L. On the left-hand wall, it has a temperature of $ T_1 $, and on the right-hand wall, it has a temp of $ T_2 $. Notice that the temperature gradient through the wall is linear. That is because the thermal conductivity is constant. This allows us to quickly solve for the convection heat flux!
Now to calculate the heat rate (in $ W $), we would need to multiply the heat flux by area; $ q_x = q"_x A $!
Now let's talk about convection. Convection is heat transfer due to a moving fluid (air, water, etc.). This is from advection, the combination of fluid dynamics and conduction. I like to think of conduction as standing still in a cold room with no moving air. The heat from your skin will warm the air molecules that make contact with your body, whereas if we were to add a space heater in the cold room, then we not only have conduction at play, but moving fluid too.
In the previous section, we learned how to solve for the conduction heat flux in a one-dimensional, steady conduction with constant thermal conductivity. Well for convection, it has its own equation for calculating heat flux:
[ 3 ]
$$ q_{conv}^" = h(T_s - T_{\infty}) $$
We can interpret Eq. (3) like so: the convection heat transfer coefficient, $ h $ in ($ \frac{W}{K•m^2} $), is the effectiveness of how heat transfers to the moving fluid. Now as $ h $ increases, so will the amount of heat transferred. This is multiplied by the difference between the surface temperature, $ T_s $, and the freestream temperature, $ T_{\infty} $. You can imagine, the greater this temperature difference is, the more heat is going to want to move from the higher temperature source to the lower, hence increasing $ q_{conv}^" $.
Convection has a few different types of convection.
The convection heat transfer coefficient increases from Natural Convection to Forced Convection to Boiling or Condensation.
What is interesting is what happens at the surface of an object where convection occurs. There is what's called a no-slip condition, where the velocity of the air is zero at the surface and increases in magnitude as you get further from the surface until the fluid's freestream velocity is reached. What is left behind is a boundary layer describing the velocity and temperature distributions. This can be visualized in Figure 2.
Fig. 2 - Visualization of no-slip condition at object's surface with boundary layers.
Notice when we talk about heat flux at a surface, the heat flux is normal to the surface.
The third type of heat transfer that we will talk about is radiation. Radiation is often referred to as thermal energy and it is transferred via electromagnetic (EM) waves. EM waves have different regions based on their wavelength.
Radiation involves emission E in $ ( \frac{W}{m^2} ) $ from the surface and can also involve the absorption of radiation that interacts with its surroundings. This is called irradiation. We can write the outflow due to emission as
[ 4 ]
$$ E = \epsilon E_b = \epsilon \sigma T_s^4 $$
Here, $ E $ is the emissive power in $ ( \frac{W}{m^2} ) $, $ \sigma $ is the surface emissivity $ ( 0 \le \epsilon \le 1 ) $, $ E_b $ is the emissive power of a blackbody (perfect emitter), and $ \sigma $ is the Stefan-Boltzmann constant ( $ 5.67E-8 \frac{W}{m^2•K^4} $ ). Radiation energy can also be absorbed due to irradiation.
[ 5 ]
$$ G_{abs} = \alpha G $$
$ G_{abs} $ is the absorbed incident radiation in $ ( \frac{W}{m^2} ) $, $ \alpha $ is the surface absorptivity $ ( 0 \le \alpha \le 1 ) $, and $ G $ is the irradiation in $ ( \frac{W}{m^2} ) $.
If you have a small gray surface, then $ \alpha = \epsilon $ with large surroundings. In this case, we can solve for the radiation heat flux.
[ 6 ]
$$ q_{rad}^" = \epsilon E_b T_s - \alpha G = \epsilon \sigma (T_s^4 - T_{sur}^4) $$
Eq. (6) can be rewritten in a different way to relate the radiation heat transfer coefficient, $ h_r $ in $ ( \frac{W}{K•m^2} ) $.
[ 7 ]
$$ q_{rad}^" = h_r (T_s - T_{sur}) $$
$$ h_r = \epsilon \sigma (T_s + T_{sur})(T_s^2 + T_{sur}^2) $$
Now, let's say we combined multiple heat transfer types such as convection and radiation, then the combined heat flux would be
$$ q^" = q_{conv}^" + q_{rad}^" = h(T_s - T_{\infty}) + h_r (T_s - T_{sur}) $$
We now have a better understanding of the 3 types of heat transfer and how these values are computed!