Nothing is perfectly efficient, not even thermal devices that operate on heat, even in ideal cases. The only exceptions are when you are maximizing heat generation or moving heat around (which can actually exceed 100% efficiency). From an engineering perspective, those device inefficiencies result in heat generation. Heat can also come from the external environment, like if you happen to be piloting a subterrene deep down in the depths of the Earth, or less fantastically, when you are being warmed by the sun's rays.
As said in the article about Heat, heat is a flow of entropy with an associated energy, and neither entropy nor energy can be destroyed. Therefore, the heat must be moved somewhere else, or kept in a place where it won't bother you (insulation - though in practice, nothing is a perfect insulator, and so the heat transfer will occur, just on a very slow timescale).
The cause of it all
Most, if not all losses that generate heat, can be traced down to three major things: Friction, electrical resistance, and energy conversion. It's impractical and pretty much nearly impossible to eliminate friction, and unless you have superconductors with zero electrical resistance, we're stuck with plain old copper wiring and semiconductors. As for energy conversion, it turns out that there is only one form of energy conversion which we can make 100% efficient: directly generating heat. We can achieve this with a variety of means, such as with electrical resistors; indeed, your typical home electric heater is a good example of such a perfectly efficient device.
The most famous example of how energy conversion is lossy is Carnot's ideal heat engine . From a thermodynamical perspective, the reason that fundamental inefficiencies and waste heat exist, comes down to the impossibility of absolute zero, due to quantum uncertainty. Thanks so much, Heisenberg!
Generally speaking, a heat pump is any device that makes use of work to transfer heat from a cool space, to a hot space. Heat always moves from hot to cold, allowing the system to reach thermal equilibrium - this is the second law of thermodynamics - but we can keep it from reaching equilibrium so long we provide a continuous supply of work.
For example, let's say we have a spaceship with an engine generating tons of waste heat. We want to reject this heat out into space obviously, but let's say due to our design requirements that the radiator has to run hotter than the engine. Because of the second law of thermodynamics, we can only do this by pumping heat from the engine to the radiator.
With heat generation, we can only achieve a maximum efficiency of 100%. We measure their, and heat pumps' performance by the ratio of useful heating or cooling provided to work (energy) required - a "Coefficient of Performance" (CoP) - 100% efficiency is represented by a CoP of 1. It turns out that heat pumps are far more efficient than such heat generators, with everyday air conditioners achieving CoPs of 2.5 to 3 (roughly corresponding to 250% or 300% efficiency). This is because heat pumps can bring in additional heat from other sources, rather than just converting work to heat as with say, an electrical resistor.
Thermodynamically, heat pumps are modelled using what's known as "heat pump cycles", or refrigeration cycles. There are many such different ones:
Vapor absorption cycle
Reversed Carnot cycle
Heat rejection can refer to the entire system of moving heat around from the heat source to the desired location. In the purview of this article however, we are referring to only the systems that transport heat from a given machine to its surroundings. If a machine has a heat source inside of it, heat rejection can refer to the whole system of coolant lines, heat pumps etc and the final system that interfaces with the surroundings, e.g. a radiator. For this section, we will be focusing on only these final, interfacing systems here.
E.g. the ocean, or the atmosphere.
Convective cooling makes use of aritifically induced (forced) convection as the primary form of heat transport. The easiest and most common way of doing this in any fluid is through pumps, or propellers.
Evaporative cooling makes use of a fluid's enthalpy of vaporization (roughly, the amount of heat energy needed to vaporize some unit of fluid). This can be significantly more energy-efficient than refrigeration. A common example of evaporative cooling in action is human sweat: on average, we reject 2257 kilojoules of heat for every liter of sweat vaporized (at 35 degrees Celsius).
However, evaporative cooling only works within an atmosphere - particularly a dry atmosphere. This is because some amount of pressure and temperature is needed for most vapors to exist. Traditional evaporative cooling fails when the relative humidity is too high (the ratio of the current partial pressure of vapor in a given volume of air to this volume's maximum capacity (saturation vapor pressure)).
Still, it is possible to make use of evaporative cooling even there, through indirect cycles. The best working fluid for evaporative coolers is water, as it has an incredibly high enthalpy of vaporization (see the section on phase transitions for that).
Outer space, baby!
Only two out of the four ways of heat transfer are present in vacuum: radiation, and advection. This is because vacuum is literally the lack of material - and without such to serve as a medium in which to conduct heat, both conduction and convection are not possible at all.
First of all, why radiators? As you might suspect, we're going to a lot of trouble with how long the section ends up - and for what? Mass is at a premium for spacecraft. So, it's easy to see why radiators can be so attractive: we don't need to bring any extra mass that we would have to for advective cooling.
The design of radiators revolves around the emission of heat as light. To emit as much light as possible per unit mass, and also per unit volume, we must maximize the surface area of the radiator. Naturally, fractals seem like the best option. However, when considering fractal shapes (such as a Menger sponge) we immediately run into two problems. First of all, they're clearly impractical and expensive to make. Second, light emitted in many places will be reabsorbed in other places, as these surfaces surround each other in a Menger sponge except for the outside; obviously, this is bad for the radiator's own efficiency.
With these two principles in mind (maximization of surface area, and avoidance of self-reflection, or self-illumination) we find that the best shape for a radiator is a flat polygon. Further extending the principle of avoidance of self-illumination from the individual radiator to radiators (plural), we find that it is desirable to place radiators in locations where their emitted light will not reach other radiators. Likewise, considering the entire spacecraft as a whole, it's also desirable that the radiator will not shine too much of its light on the spacecraft's body.
This is why many spacecraft designs have a "coplanar" layout with two radiators sharing a plane going through one of the spacecraft's axes of symmetry.
[Section about the benefits and tradeoffs of more radiators]
The Stefan-Boltzmann Law, discovered in the 1880s - describes the dynamics of radiation here:
- where is the radiant power - i.e. amount of heat energy rejected per unit time.
- where is the radiating surface area.
- where is the emissivity of the radiator's material.
- where is the Stefan-Boltzmann constant of proportionality.
- where is the temperature of the radiator.
Technically, this equation is a derived form of the Stefan Boltzmann Law - instead breaking up intensity into power and surface area. Put simply, the Stefan Boltzmann Law is the relationship of a matter's own temperature to the intensity of thermal radiation it emits (amount of energy per unit area per unit time emitted).
The best radiator would be what's known as an ideal black body - something which absorbs 100% of all light. Since physics is reversible, it also means that this ideal black body has a perfect emissivity - i.e. it emits 100% of its own thermal energy as light. Emissivity is just how close a given material is to being this ideal blackbody - the ratio of the measured radiance to the blackbody's theoretical radiance. Nothing in the real world can be an ideal blackbody. Stars and other astrophysical phenomena, do however, very closely approximate one.
It turns out that increasing the temperature of the radiator will increase the radiant emittance (intensity), all other things equal. This allows us to decrease the surface area of the radiator, which has some very good engineering benefits: the mass of the radiator (since mass is at such a premium for rockets) goes down, and for a combat spacecraft the radiator is a much smaller target. We also want as high of an emissivity we can get, so blacker materials are more preferable. Nevertheless, it's important to note that by blacker it's meant that the material is "black" over the entire electromagnetic spectrum; so materials which appear highly reflective (like ice) in the visible spectrum, a silver of the entire EM spectrum in of itself, may actually have very high emissivities (0.97-0.99)!
Classic radiator designs, are made of solid materials and are evenly hot. We can make it evenly hot just by pumping heat around in a crisscross lattice of channels throughout the radiator.
A comprehensive table of emissivities is given here:
Dusty plasma radiators
Open cycle (advective) cooling
Oh, what a bother! All that radiation, so much work!
We simply do it like God intended: throw the heat out. Open cycle cooling systems (advectors) can make use of the engine's exhaust itself. However, this only works when the engine has enough thrust (propellant flow) to sustain the required amount of heat rejection: to keep open cycle cooling going when that is no longer possible, one can simply vent hot fluid out into space using internal reserves of some coolant fluid.
As mentioned above, open cycle cooling is far more wasteful when it comes to mass than radiators, hence why they're not as common of as a heat rejection system for spacecraft (where again, mass is at a premium). We want to be very efficient with our open cycle cooling. We must squeeze as much heat we can into every kilogram we throw out. It turns out that water once again, is the best kind of working fluid for this: it has a very high specific heat capacity (amount of heat energy needed to raise an unit mass by one degree of temperature). We can also make use of phase transitions (such as its enthalpy of fusion (melting) and enthalpy of vaporization).
Open cycle cooling is very attractive for combat spacecraft, where huge radiators are a huge liability. In fact, open cycle cooling reduces the heat rejection apparatus down to a hole, or perhaps a nozzle - where the hot fluid is expelled. No great love lost with shooting mere hot air, unlike classic radiators - where shooting holes can spring leaks and cause all manner of unpleasant damage!
Because of this, combat spacecraft likely make use of a mix of both radiators and open cycle cooling: when they enter combat, they fold in their radiators or however so, and switch to open cycle cooling for the duration of their combat. Likewise, they can also use heat sinks.
A heat sink is fundamentally a delaying action in the war against heat. You dump all of your heat into it during a period of intense usage, and then plan to radiate / throw it away later. This is especially useful for applications that emanate very large pulses of heat, where radiators and advectors would struggle to keep up - like a laser weapons system, or a spacecraft during a particularly intense combat.
As mentioned above in the section about advection, heat sinks and advectors can be described using heat capacity. Given that the unit of specific heat capacity is heat energy divided by mass divided by temperature, one can devise equations to solve for the mass of the heat sink / amount of material to advect, or the amount of heat energy that the sink/advector can absorb, and so on. Heat sinks will be limited by your design considerations: do you want a solid heat sink, or are you willing to let it melt? At which point does the heat sink pose a liability to the spaceship itself?
It's important to note that specific heat capacity also changes with the state of matter, and even with temperature. Furthermore, for absorbing intense pulses very fast, the heat sink should also have a good thermal conductivity. A slight wrench in this consideration however, is since that the heat sink will be surrounded by other parts and machinery of the spacecraft - it will also be dissipating heat to these parts. This can be solved with judicious application of insulation and design.
Advectors, of course, have looser constraints - if you, oops, end up with plasma - you can simply throw it out.
As it turns out, heating a material up to its critical temperature for a phase transition (say, solid to liquid state) will not always make it melt. An ice cube may be at a temperature of 0 Celsius, but without additional input of heat energy to make it melt, the ice cube will remain solid. This is called the enthalpy of a phase transition (even though enthalpy is a different matter in of itself, being the sum of a system's internal energy and the product of its pressure and volume). It is also called the latent heat, because during the phase transition, all heat energy goes toward making it happen - there is no change in temperature.
Phase transition enthalpies can be a powerful way to increase the total capacity of your sink/advector, and can be a big reason why some materials are desirable, even if they have a poor heat capacity; these enthalpies are also the working principle behind evaporative coolers.
Some potential suggestions for heat sink / advector materials:
Note: If temperature/phase is not specified, assume that it is at standard temperature and pressure conditions. All values are given at constant pressure (isobaric).
|Material||Specific Heat Capacity
|Volumetric Heat Capacity [Cal 2]
|Specific Enthalpy of Fusion/Sublimation[Cal 3]
|Specific Enthalpy of Vaporization[Cal 4]
|Specific Ionization Energy |
|Hydrogen gas||14.3||0.5||58.04||222.9 or 446.1||1302.88|
|Water (25 °C)||4.1813||4.1796||333.55||2257||67.589|
|Water, ice (-10 °C)||2.05||1.938||333.55||2257||67.589|
|Water, steam (100 °C)||2.08||0.00125[C 1]||333.55||2257||67.589|
|Lithium, liquid (181 °C)||4.379||2.26||432.3||21030||74.96|
|Graphite||0.71||1.534||59700 (Ent. of Sublim.)||59700(Ent. of Sublim.)||90.5|
|Diamond||0.5091||1.782||59530 (Ent. of Sublim.)||59350(Ent. of Sublim.)||90.5|
|Steel (stainless)||0.466||3.756||267.776||7406||14.421[C 3]|
An important consideration is that the specific ionization energy refers primarily to the first ionization energy (i.e. the energy needed to strip the first electron off the atom.) Second, third, and other ionization energies are not given.
- There are many routes to calculate specific heat capacity. However, if the molar heat capacity for your desired material is available, you can use the following formula:
- where is the isobaric specific heat capacity
- where is the isobaric molar heat capacity
- where is the molar mass
- Like specific heat capacity, there are many routes. If you desire to use a similar formula to the one for , you can switch molar mass out for molar volume .
- Molar-specific conversion formulas are also possible here.
Notably Calculated Figures:
- 2.08 kJ/(kg⋅K) ⋅ 0.6 kg/m3
Converted directly from specific heat capacity using density.
- (78⋅(1402 kJ/mol / 28.041 g/mol)+21⋅(1313.9 kJ/mol / 31.998 g/mol)+1⋅(1520.6 kJ/mol / 39.948 g/mol))/100
Not very sure about the correctness here. There may be some emergent properties that could decrease or raise the ionization energy.
For the analysis, it was decided to go with 78% nitrogen, 21% oxygen, 1% argon - and then a weighted average for the ionization energy.
- (99⋅(762.5 kJ/mol / 55.85 g/mol) + 1⋅(1086.5 kJ/mol / 12.011 g/mol))/100
Probably incorrect, like the treatment of air. Analysis: 99% iron, 1% carbon.
The primary and theoretical purpose of insulation is to stop the flow of heat transfer from one object to another. However, in practice, we can only achieve a reduction of heat transfer, since the second law of thermodynamics, as well as other quantum phenomena, forbid what's called a "thermal superinsulator" - a material which would be able to completely stop heat transfer.
Heat, as we all know, is transferred through four mechanisms - conduction, convection, advection and radiation. We can stop heat conduction from happening, and convection by extension since it is contingent on conduction, if we put a vacuum gap for our superinsulator. With no work is being exerted on both of the surfaces surrounding the gap, this eliminates advection. Therefore, the only mechanism left by which we can transfer heat is through radiation.
For a superinsulator to eliminate this mechanism, it would need to suppress all possible oscillations that could cause radiation - which is impossible, due to Heisenberg's uncertainty principle. The second law of thermodynamic strikes again. Even theoretically speaking, some, extremely minute, heat conduction will occur due to spontaneous offgassing and the impossibility of a perfect vacuum (due to quantum fluctuations). This is all aside from the fact that pulling a vacuum strong enough to be dominated by quantum fluctuations would also be ridiculously difficult, and certainly impossible with modern technology.
So conventional insulators can hope for as low as a thermal conductivity as current materials sciences can afford; vacuum insulators are limited by the emissivity of the two surfaces on each side of the vacuum gap. More insulation can be achieved with multiple layers; there's also the phenomenon of two different materials being layered together having an insulating effect of its own, as heat transfer drops across the material transition/discontinuity.
The catch with vacuum insulators is that they are generally more expensive and more inflexible in terms of shapes - they usually take the shape of panels, unlike other insulators which are more affordable and can be easily worked into any shape. However, in the future with cheaper energy and more advanced manufacturing techniques, vacuum insulators could be more widespread.
Authors: Qalqulserut, Rocketman1999
Stefan Boltzmann Law
Reference for volumetric heat capacity of hydrogen and helium.
- This latter figure is for parahydrogen, a spin isomer of hydrogen. Liquid hydrogen can consist of various ratios of orthohydrogen and parahydrogen.
- Converted from 64 cal/g to kj/kg
Reference for both steel's specific enthalpy of fusion and vaporization.
- Converted from 1770 cal/g to kj/kg