Header image: Firefly radiators. The orange glow comes from the heat at 2400 K. The radiators are used to cool the ship structure, that is heated by neutron and x-ray radiation from the drive pinch (the narrow white line at the bottom of the image). A liquid metal circulates through the structure and the radiators, transferring heat gained in the structure and releasing it in the radiators (ML)

Introduction

A fusion Starship concentrates the power and energy of a continental electrical grid into a vehicle the size of a small building. Inevitably, the handling of all that energy cannot be perfect. Inefficiencies will exist, and these will create heat, that may exceed the capacity of the materials that make up the starship and will need to be removed. Some of the critical elements of the starship need to be kept quite cold, such as the fuel and the superconducting parts in the power and communication systems. These two functions, removing excess heat and keeping things cool are the purpose of the thermal controls systems.

Practically all the elements on the ship need some form of cooling. The drive, the nozzle(s), the electronics, the power systems, the fuel tanks. The best strategy is always to avoid heating altogether. Therefore, the main energy source, the fusion drive, is often open to space, to allow energy losses due to inefficiencies to escape without creating any heat at all. Other areas of the starship are protected by heat shields. These either reflect the energy or have cooling loops that carry excess heat away from equipment and towards radiators, where the heat can be radiated away into space.

Thermal shields

For Firefly, the radiation shield is also the structure. This design has a high radiation load, and requires very powerful and efficient cooling. A liquid metal evaporates in the hollow structure, capturing heat in its phase change. The gas moves to the radiators, where it cools and condenses, expelling the heat. The liquid metal is pumped back to the main structure where the cycle begins again.

In a certain sense, the drive mostly cools itself. The stupendous heat created by the fusion reaction is absorbed by the fusion plasma particles and transformed into pressure in the nozzle, that pushes the ship, and then into velocity that carries the particles away, bringing along most of the heat.

All of the drives types have radiations losses, that escape the drive area and heat the structure of the ship. This heat must be removed to be released into space at the radiators. Firefly in particular, has losses that are greater than the power of the drive itself, in the order of 250 MegaWatts per square meter. This will destroy the structure in a few minutes if the heat is not removed. As a comparaison, ITER is expected to have a heat flow of 20 MegaWatts per square meter at the thermal blanket₍₁₎.

In the original Daedalus design, the drive nozzle served a dual purpose, containing the drive plume and radiating away excess heat. In fact, for the first stage, the size of the nozzle, 50m in diameter, was a consequence of the maximum allowable temperature of the material used, Zirconium (r), and of the heat induced by the radiation from the fusion reaction and eddy currents from induced magnetism. Simple conduction and radiation systems were adequate for the 9GW thermal load of the first stage, and the 1 GW of the second stage. For recent designs, the estimated radiation loads have increased. The main changes comes from new evaluations of the fusion reaction itself. Modeling done since the seventies has shown that for all types of fusion drives, including all types of aneutronic fusion reactions, radiation levels, in the form of Bremsstrahlung x-ray radiation and/or of high energy neutrons will be higher than what the Daedalus designers had planned for. In fact, for all of the designs presented here, the closed nozzle such as used by Daedalus couldn't survive the radiation load.

So the designs have mostly switched to an open nozzle, using magnetic containment that is transparent to radiation, generated by superconducting magnets protected from the radiation by an actively cooled heat shield. As the rate of heat gain has increased, more powerful cooling systems are required, with higher heat removal rates. Mass flow and phase change systems are needed to transfer the heat. The mass flow can be a circulating gas, liquid or a mixture of both, that includes phase changes back and forth between liquids and gases. The highest possible heat transfer rates are achieved using phase changing liquid metals. See table 1 for some examples.

Cooling with fuel

The question is often asked: Can the fuel itself be used to cool the drive, as for chemical rocket nozzles? The answer for fusion drives is no. The fuel flow in a fusion drive is very small, in the order of grams per second. For a mass flow of 100 g/s of deuterium, and even if the temperature of the fuel was elevated by 1000 degrees before it entered the fusion reaction, the fuel could only remove 500 kiloWatts, a tiny quantity compared to the GigaWatts of heat that needs to be removed

Q=energy gain in fuel(W)

Q=m*CP*dt

Q=0,1kg/s*5.1*1000=510 kW

m= Mass flow of fuel (kg/s)

Cp= Specific heat of fuel (KJ/kgK)

dt=Temperature change of fuel (K)

Cooling the magnetic coils

Magnetic coil shielding

A superconducting magnetic coil and its shielding. In the shield, the liquid metal coolant absorbs neutrons, the high density metal (probably tungsten) absorbs x-rays, heating up considerably. The coolant evaporates and the gas carries away the heat to be radiated into space at the radiators. The multilayer insulation protects the superconductors from the thermal radiation from the hot shield, and the liquid nitrogen coolant removes any leftover heat, to be radiated away at low temperature radiators. In the image, the fusion reaction is to the right.

To create the magnetic fields, superconducting magnets are required. These must be kept at very low temperatures to operate, and the materials that compose them are sensitive to neutron damage, eventually losing their superconductive properties under heavy neutron bombardment.

A two tiered shielding system is required for the superconducting magnets:

The first shield absorbs the neutrons and x-rays, turning their energy into heat. The shield is filled with FLIBE, lithium or beryllium, with a tungsten backing plate. All the absorbed energy is transferred by a circulating coolant to remote radiators, where it is radiated out into space. However, to be efficient this first shield must operate at high temperatures, and will radiate in infrared, and in some cases in optical frequencies as well.

The second shield is composed of multilayer insulation that can reflect this radiation, reducing the heat load of the superconductors to the minimum. The leftover heat gain can be removed by a liquid nitrogen circuit, that moves the heat to low temperature radiators located far from the heat of the main drive. Higher temperature superconductors allow the designers to operate the cooling loop at higher temperatures, and therefore make its components lighter, but cannot cancel the heat gain.

The liquid nitrogen circuit removes heat by mass flow as per equation (5) , and the multilayer insulation functions as per equation (19). The nitrogen transfers the heat gain to a cryocooler that rejects the heat by radiation. With a shield temperature of 2500 K and 50 layers, 26 KW can be removed from a 70 K superconducting magnet using 8 kg/s of liquid nitrogen with a 10 K temperature difference and 124 kW of power. The resulting radiator has an area of 230 m2 at 400 K and dissipates 150 kW. The cooling system might weigh about 10 000 kg.

The main challenge will be designing the interfaces between the shield and the structure, and developing thin materials that can withstand the temperatures involved.

Secondary systems cooling

Whenever possible, a starship will use superconducting power systems to reduce losses and lower mass. Superconducting motors, magnets, transformers and conductors are all much lighter than their standard conductor counterparts and have much lower losses. Since the ship is operating most of the time in deep space, the superconductors are relatively easy to cool. Modern high temperature superconductors can be cooled using a nitrogen coolant circulated in a loop. Ammonia is an alternative for future higher temperature superconductors.

While the ship is charging up in preparation to lighting the drive, or while it is in coast mode, a conventional fission reactor provides the required power. Liquid metal cooled breeder reactors are a well developed technology that could be used for this purpose. The liquid metal can be sent directly to radiators, or heat exchangers can be used to heat a secondary coolant, if circulating the liquid metal is problematic. During coast, the ship can actually get quite cold, and the heat from the secondary power reactor will be required to ensure ship systems do not go below their lower temperature limits.

The emitters and amplifiers used in the communication system with the Earth produce quite a bit of heat, and the superconducting magnets that are used to modulate the emitters need cooling. Approximately 60% of the power required will be lost as heat, See chapter 11, Communications, for details.

Fuel tank cooling

The fuel tanks need to be kept cool, down to 4K in the case of He3 tanks. Since this is colder the the background temperature of space, refrigeration is required for helium tanks to move the heat up to a temperature that allows radiators to function.

All refrigeration processes use an external energy source to provide the work that moves the heat from one location to another. The refrigeration system operates against the heat gradient, moving heat “upslope” from a cold space to a hotter space. It has to work to get things done, and that work also turns into heat, that adds itself to the energy that has to be dissipated. In most refrigeration systems in common use today, the work required to move the heat is about 30% of the value of the heat itself. But if the slope is very high, going from a very cold place to a very hot place, the work can add more heat than the actual heat that is removed.

The Carnot cycle is the theoretical description of the refrigeration cycle. It relates the work required to the temperatures of the hot and cold states of a system. The performance of the system is given by the coefficient of performance (COP).

Firefly fuel tank. The radiator is dark grey. A compressor located in the radiator system extracts heat from the tank and rejects it into the radiator using a Carnot cycle compression refrigeration system.

Tank cooling example

COP=Coefficient of performance

For the cooling system refrigerating a deuterium tank at 18K, and radiating using radiators at 288K, or just about 12C, the COP = 18/(288-18) = 0.067. Again, because of imperfections in the cooling system, the actual COP is about half the theoretical one, so the COP will be 0.034. The work required to remove 1 kW is W= 1 kW/0.034 = 29 kW or 29 times the energy removed, and the radiator will have to dissipate 29 kW + 1 kW = 30 kW.

COP= T_L/(T_H-T_L) and Q=Qc/COP

T_L= Temperature, cold system (K)

T_H=Temperature, hot system (K)

Q= External power required (W) (J/s)

Qc= Power extracted, cold system (J/s)

Q=Radiation(W)

Q=Be*A*(Ts⁴-Ta⁴ ) For the tank illustrated here, the radiator area is about 24m². At 288 K, the maximum cooling the radiator can provide is ) which work out to about 8 kW. So the heat gain must be kept below 8/40 = 250 Watts. Therefore, we must design the ship so that a minimum amount of radiation coming from the drive and the main radiators penetrates to the tank, less than 250 Watts over the entire tank surface

Be= Stefan Boltzman constant

A= Radiatior area (m2)

Ts-Ta = surface temperature - space temperature (K)

The first line of protection is the main vehicle shield, that absorbs most of the neutrons and has been described above. However, it is practically impossible to hide the tanks from the thermal radiation coming from the drive radiators, and even if we could place a shield, it would itself get hot, and radiate towards the tanks anyway. The dust shield at the head of the ship also heats up, and the tanks must be protected from its heat as well. To reduce refrigeration, we need to provide some insulation.

Insulation

In space, insulation is provided by multilayer reflective insulation, or simply Multilayer insulation. A technology that has been used for decades on satellites and in cryogenic equipment (equipment operating below -180C), and is quite mature. It can allow very hot elements to coexist with cold ones by limiting radiative heat transfer. See equation 19.

In multilayer insulation, the heat transfer is simply a function of the number of layers. Add enough layers, and you can insulate any range of temperatures one from the other. The only limitation is the melting points of the layer materials. In practice, the insulation layers need to be separated and kept in place by spacers, who conduct heat between the layers and reduce the overall effectiveness, so a safety factor should be added to the design.

Refrigeration cycles

The Carnot equation is a purely theoretical formula. The most common system that can be used to implement it is the vapor compression cycle. Introduced in the late 1850s, it is still the most energy efficient system today, despite its abundance of piping and rotating parts.

The cycle starts with a compressor, that puts a gas in motion using energy from an external source and compresses it to a high pressure, which heats the gas. The hot gas goes through the condenser, a heat exchanger that removes enough heat that the gas turns into a liquid. The phase change from gas to liquid liberates heat that is radiated/conducted away, out of the system. The liquid is then pushed through a pipe to an expansion valve, in which the pressure is reduced dramatically. The liquid turn back into a gas, and this phase change absorbs heat at the evaporator. The hot gas then returns to the compressor and completes the cycle. The compression moves the gas, and requires an external work input. The phase change moves the heat. By using piping to make the two phases changes happen at two different places, the heat is transferred. The greater the temperature difference between the cold side and the hot side, the more compression is required. Eventually, the heat from the compression work heat becomes more important than the transferred heat, but overall the cold side is cooled anyway.

There are friction losses, irreversibilities in compression, inefficiencies in the compressor that all reduce the COP considerably from the ideal values of Carnot’s equation. Large vapor compression units can reach 60% of the efficiency of the ideal Carnot Cycle, but small units are considerably less efficient. So there is room for improvement.

Weight of cryogenic cooling systems

A typical existing cooling system() weighs 1 000 kg for 4 kW of cooling at 70K, using 45 kW of electric power and rejecting 49 kW of heat. It operates at 3 000 kPa, or 450 psi. Space optimized system would be considerably lighter, in the order of 50 kg per kW of cooling for the superconductors. However, considerable technological development would be required to reach these mass numbers. At the lower temperature required for fuel tanks, a similar 1000 kg unit will only produce 400 W of cooling for the same 45kW of power. A value of about 500 kg/kW should be achievable, so to cool a fuel tank by 280 watts, a 140 kg system would be needed.

Phase change cooling

The following table lists some interesting materials for spaceship cooling using the vaporization phase change or the mass flow. It highlights the fact that, in most cases, the vaporization phase change transports much more energy than temperature change in the fluid phase.

From Table 1 we can see that most coolants need much less mass flow to carry 100 GW through vaporization phase change than by heating of the liquid or gas. Taking silver as an example: As a liquid, it requires 833 tonnes per seconds at 500°C of temperature difference. But by changing from a liquid to a gas, it requires 42 tonnes per second, practically 20 times less coolant to move the same amount of energy.

Although the equation is simple, the physical phenomenon of boiling isn’t. The transition from a liquid, to bubbles, then to a bubbly froth and finally into a gas is accompanied by a large change in volume, and is influenced by the pressures in the boiling area. Fluids are good heat conductors but gases are bad ones, so the vaporisation must happen at the right point, or else the equipment that is supposed to be cooled may melt, despite the presence of the coolant at the correct temperature. In a phase change system, the radiation shield effectively becomes a boiler, where neutron and x-ray radiation play the role of the burner, and the liquid metal the role of water and steam. Depending on the design and the materials used, the radiation may heat the liquid metal directly through neutron capture, or heat an intermediary material that heats the fluid through conduction. Radiator designs are limited by the maximum temperature that the pipe walls, as well as the pumps and compressors, can withstand.

One important point in using the vaporization phase change to transmit the cooling power is that the whole evaporation condensation cycle takes place at practically the same temperature. This means that the whole radiator surface will be approximately at the phase change temperature. This is a significant gain in radiator efficiency, reducing their area and therefore their mass, by 30%.

For example, let’s suppose the coolant leaves the shield at a temperature is 2000 K. A radiator with a helium coolant and with 500 C of temperature difference would be, at best, at an average temperature of 2000 - 500/2 = 1750K. The phase change radiator using lithium could be at practically 2000K for its whole surface; let’s say 1950K.

Pressure

At what pressure level should we expect our system to operate?

Table 1 is based on atmospheric pressure, but that may not the best choice. For lithium, the radiator temperature is perhaps a little low. To increase the temperature and use phase change at this higher temperature, you need to increase the pressure, or else the fluid will boil too early. However, if we operate at higher pressures, the phase change energy will go down. Eventually, at high enough pressures, you don't get a phase change at all, the metal remains liquid despite the heat added. This is the area of supercritical fluids. A higher boiling point is interesting, since it means the radiators will be operating at higher temperature, and therefore be able to handle more power.

The perfect gas laws can be used to determine the pressure we would like to operate at, in order to get a certain density of gas. Higher pressures will increase the boiling point and reduce the energy in the phase change. Clapeyron’s Equation can be used to determine the new boiling point, and then Watson’s Equation can be used to estimate the phase change energy at these new values. Finding the optimal solution will require an iterative process (that the mathematically inclined are invited to solve for the optimal point), as we choose between a higher pressure that gives us a higher density and smaller piping, and a lower pressure that maximizes the phase change energy.

For example, as a first approximation, we might want to choose to operate our Lithium system at 2000°K, for maximum radiation power.

We can use Clapeyron’s Equation to confirm at what pressure the lithium will vaporize:

3) P₂=P_n*e^V_e/R(1/T_n-1/T₂)

Where:

P₂=new pressure at boiling point (kPa)

P_n=atmospheric pressure = 101 (kPa)

V_e= vaporisation energy at standard pressure = 146 (kJ/mol) for lithium

R=perfect gas constant = 8.3144 (J/°K*mol)

T_n=boiling temperature at standard pressure = 1600 (°K)

T₂= New temperature = 2000 (°K)

Filling in values for our example:

P₂= 101*e^146,000/8.314*(1/2000- 1/1600 = 900 kPa

This will be the operating pressure of the system.

To determine the density at this pressure, we used the Perfect Gas Law:

4) d=A_w*P / R*T

Where:

d=density (kg/m³)

R=perfect gas constant = 8.314 (J/°K*mol)

A_w=atomic weight = 6.9 g/mol

P=pressure (kPa) = 2000 kPa

T=temperature (°K) = 2000 °K

6.9*900 / 8.314*2000 = 0.37 kg/m³

Finally, using Watson’s Equation we can find the new heat of vaporization:

5) Ve₂= Ve *((Tc - T2)/(Tc - Tn))^0.38

Where:

Ve₂=vaporisation energy at new pressure (kJ/kg)

Ve= standard vaporisation energy (J/kg)= 146,000 (kJ/mol)/6.9 = 21159 kJ/kg

Tn=standard pressure boiling point (°K) = 1600 °K

T2= New temperature (°K) = 2000 °K

Tc= temperature at critical point (°K) = 3220 °K for lithium

Again using the values from our example:

Ve₂ =21159*(3220-2000/3220-1600)^0.38 = 19,000 kJ/kg

So the phase change at this pressure carries 10% less energy, but the radiators should have a much greater gain in efficiency.

Referring back to Equation 1), for 100 GW (or 100e9 J/s) of power:

100e9 J/s ÷ 19e6 J/kg = 5,300 kg/s.

And 5,300 kg/s ÷ 0.37 kg/m³ = 14,300 m³/s.

These reduced mass flows will allow the designer to use slower velocities in the piping, thereby reducing both the pumping power required, and the wear on the piping.

The main problem with all of these good numbers is that lithium vapor at 2000 °K is highly reactive. It will be a challenge to develop materials able to resist it for the many years of operation required. If it were possible to use Beryllium as a heat transfer medium, the mass flow would go down even further, to about than 3000 kg/s. However, the piping and equipment would then have to withstand temperatures in the order of 2700-2800 K. Firefly version has chosen Beryllium, while most of the other design use lithium or helium.

Lithium as a coolant/radiation shield has the interesting property that when it intercepts neutrons, it may produce Tritium as a by product. For the drives that can use tritium in the fusion reaction, the coolant can breed new fuel.

Conduction

Conduction plays a part in all cooling systems, as the heat eventually has to traverse materials to reach a surface to radiate away from.

Thermal conductivity, the rate of conduction, is an inherent property of materials. Graphene has the highest thermal conductivity, but only in the direction of the sheet. Diamond has the highest thermal conductivity of ‘standard’ materials. Liquids are usually less conductive than solids, and gases have poor thermal conductivity.

Fluid are bad conductors compared to solids

A surface temperature of 2200 K (Tout), with an emissivity of 0.9, corresponds to a power of about 1000 kW per m2. If we suppose a thickness (x) of 1cm, or 0.01m, for a power (Q) of 1000 kW per m2, then we can find the temperature difference through the material required to ‘drive’ the conduction, see column 3 above.

So, for example, with a radiator made from tungsten pipe with a hot liquid metal coolant circulating inside, the inner surface temperature would be 2200K + 58K = 2258K. A significative difference, but not a deal breaker. If we tried to push that much power through a carbon fiber reinforced epoxy wall, it would melt long before it reached the required temperature difference.

References

(1) Dolan, Thoma J., Magnetic fusion technology, Springer, 2013.

(2) Lienhard, John H. A heat transfer textbook. Courier Corporation, 2013. An excellent free reference textbook covering all aspects of heat transfer.

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