Radiative Heat Transfer

Radiative heat transfer involves the electromagnetic emission of photons from objects. This is happening all the time, but when temperature differences are very low, radiation heat transfer is very low as well. The equation for radiation heat transfer illustrates this:

Because temperature is to the fourth power, and the Stefan-Boltzmann constant is so small (~5.7*10-8) only very large temperature differences will result in significant radiative heat transfer. 

A good example of radiative heat transfer is the way the Sun transfers heat towards us on Earth. The sun is uber hot, and its glow is due to its radiation in the visible spectrum. Extremely hot metal that glows red illustrates the same phenomenon. Turns out, radiation happens in more than just the visible spectrum as well (ultraviolet, infrared, gamma rays, etc.). Although you can’t see it, it’s important to know that it is there, and model for it accordingly when necessary. This has huge impacts on the design of Earth-orbiting satellites!

Emissivity

The best way to affect radiative heat transfer is by changing the emissivity of the surface. A “black body” or “black body radiation” is an assumption used with radiation where we assume the emissivity ε = 1. We can get close to an emissivity of 1 by adding a dark layer on the outer surface. We can also get low emissivities by using shiny or polished metals, which will lower the radiation heat transfer (this is why aluminum foil is used to keep food hot/cold!)

Engine Cooling Methods

Heat Sink Cooling

When using a heat sink cooling scheme, the main thing being leveraged is the transfer of heat through physical material. In this setup, the heat from the combustion chamber and surrounding areas get absorbed into the chamber wall in order to reach a point of equilibrium with its resting temperature. In order to do so successfully, the area of high heat must be surrounded by a significant amount of bulk material so that the heat can transfer effectively while keeping the metal below its melting point.

Inherently, this is a very unstable problem due to the fact that tracking the exact transfer of heat through a material isn’t necessarily well defined. Because of this, heat sink cooling schemes require the engine to be constructed out of materials that have the following properties:

• High Conductivity

• High Density

• High Heat Capacity

These properties combine in a value called the thermal diffusivity (𝛼): k/𝝆c [m^2/s]

The thermal diffusivity can be used in unsteady heat transfer calculations to see how the temperature evolves in time. This is extremely useful for modeling heat sink cooling methods, which often operate on lower time scales. This is because eventually, the entire object will reach thermal equilibrium, at which point you no longer have the heat transfer you need to keep the material from melting!

Although we have used this cooling method with the Helios engine, it is usually not implemented in industry unless the engine is rather small, has short firing times, and the temperature doesn’t need to be heavily regulated.

A regenerative engine cooling scheme is a system that passes the cryogenic propellant through the engine wall to cool the material before delivering it to the injector/combustion chamber. 

It is called a regenerative cooling system because we use the fuel as coolant, before injecting it into the combustion chamber. In such a system, we are mainly concerned with conductive and convective heat transfer. 

1. Convective heat transfer between the hot combustion gases and the nozzle wall.

2. Conductive heat transfer through the nozzle wall. 

3. Convective heat transfer between the nozzle wall and the coolant. 

This is illustrated in the figure below:

If we have a coolant temperature and a gas temperature, we can calculate how hot our wall will get by using the previously mentioned equations. The boundary condition is that the conductive heat flux from the gas to the wall will equal the conductive heat flux through the wall and also be equal to the heat flux from the wall to the coolant. 

Actual designs of regenerative cooling systems usually involve a series of small channels that are lined circumferentially around the combustion chamber and nozzle. 

Looking at a single channel, the coolant enters at some point along the engine profile, where it is pumped through at a high pressure and velocity. The coolant then exits the channel where it is eventually fed back into the injector, or sometimes dumped off as exhaust. There is a multitude in varying designs in regenerative cooling systems, which broadly involves:

• Choice of channel entry/exit point

• Number of channels

• Location of cooling channels along the nozzle (sometimes only part of the nozzle/chamber is cooled)

• Direction of coolant flow (exit plane -> injector plane, or vice versa)

• So much more!

The geometry of the series of channels involves a few different geometric parameters as well:

• Chamber wall (internal wall) thickness: We want to decrease this value to get better conduction, but are limited by the structural integrity and machinability of the material and structure.

• Channel Rib (fin) thickness: We want this value to be small as well so that the fins can conduct heat from the chamber wall to the coolant.

• Channel height & width: These parameters affect the cross-sectional area and aspect ratio of the channel (an aspect ratio of 1 is a square channel). To get better convective heat transfer, we want to have many small channels, as this will increase coolant velocity.

When designing regenerative cooling systems, the heat transfer coefficients are very important to keep in mind. In our case, we have a hot side and cool side HTC, and these values can be tweaked to adjust the temperature profile along the wall and boundary layers. Since the goal of engine cooling is to keep the material temperatures low, we will want our hot-side wall temperature to be as low as possible. This means that we will want to keep our hot side HTC small, and cool side HTC large. We won’t talk in-depth about heat transfer coefficients here, as the equations get quite intense, but understanding that we want hc / hg to be very large is a good first step in understanding the manipulation of these values to get a desired temperature. 

Thermal Resistance Network

A relatively simple and common way to model networks of heat transfer is to use the circuit analogy. Here, we employ the circuit law equation and replace them with variables relevant to heat transfer:

 ->

• Current is analogous to heat transfer. Flow of electrons -> flow of heat

• Voltage is analogous to temperature. Voltage drop -> temperature drop

• Resistance is analogous to thermal resistance, and the expression changes depending on the type of heat transfer

For conduction and convection, we can easily derive the equations for thermal resistance:

Now, using the circuit analogy, we can consider more complex heat transfer problems by considering different bodies and mediums to be resistors. Two pieces of material next to each other could be considered as two resistors in series, so we could find the equivalent resistance by adding the two thermal resistances (resistance add in series).

We can use thermal resistances to calculate the value of Ts , shown in the figure above. If we know T0, T∞, k, h, and A, we can find the thermal resistance of the grey material section (conduction), and the blue fluid section (convection). If we calculate the equivalent resistance, we can plug that into the heat flux equation to get heat flux, and then look at just the material section to back Ts, knowing the heat flux is constant.

Thermal resistance networks have a huge use in industry, and they are also the fundamental basis for heat transfer modeling. Imagine a complex heat transfer problem (like the figure above) where you cut up the volume or area of interest into small cubes, define a thermal resistance in each direction, and then run a fancy algorithm that can look at each cell and/or node to find the temperature mapping everywhere!