Actively Cooled Armor for Space Warships

[MatterBeam]

This is from the latest Toughsf blog post: http://toughsf.blogspot.com/2019/05/actively-cooled-armor-from-helium-to.html

Actively Cooled Armor: from Helium to Liquid Tin.

We have seen designs for long ranged particle beams and powerful lasers. Could they be the end-all, be-all of space warfare? Not if we fend off their destructive power with actively cooled armor.

Let's have a look at the different cooling solutions, from high pressure gas to liquid metal, and evaluate their relative effectiveness.

 

Armor

The armor curves away to hide the radiators from attack from the front.

 

The traditional solution for defeating directed energy weapons such as particle beams and lasers is to use solid plates of armor. The armor material would ideally have a high heat capacity so that it doesn't heat up too quickly, and an excellent melting or boiling energy.

Graphite excels in both. It is clearly superior when compared to steel or aluminium. 

 

We have written about the effectiveness of graphite when facing laser beams, and how we can use different techniques such as sloping, rotation and reflective surfaces to further increase the energy required to remove armor. 

 

However, certain techniques described in Lasers, Mirrors and Star Pyramids have limitations. Reflectivity in particular cannot be counted upon in all situations. While we might have broadband dielectric mirrors that effectively reflect a vast range of wavelengths, an enemy will eventually field lasers specifically meant to defeat them. 

They might select polarizations against which the mirror is less effective, they might use beams of wavelengths too short to reflect (usually below 200nm), or even replace lasers with particle beam weapons that ignore surface features entirely.

 

What can we do against such mirror-defeating techniques? Can we increase the effectiveness of armor even further than what was calculated in Lasers, Mirrors and Star pyramids?

 

Maximizing that energy value means you can get by with less armor and have more mass dedicated to winning tools such as propulsion or ammunition.

 

Passive armor

 

We can start with a reference to compare everything else to.

 

Passive armor is simple in design and construction. It is made to handle as much heat as possible and prevent it from leaking into the spaceship.

Graphite electrodes in an electric arc furnace.

A good example, as mentioned above, is graphite. It first needs to be heated to about 3500K before it starts being degraded. At 4000K, it turns into a gas. Between its heat capacity and vaporization energy, it takes roughly 60 MJ/kg to vaporize. A stronger carbon-based material would take an equivalent amount of energy while also being physically strong.

 

We can therefore expect graphite to handle a laser intensity of 8.5 MegaWatts per square meter for extended periods of time. This is a value calculated from the Stefan Boltzmann blackbody radiation equation, as it is the intensity required for a black surface to sit at an equilibrium temperature of 3500K.

 

For the laser beam to start digging into the carbon at an appreciable rate, it must first overcome a 14.5 MW/m^2 threshold so that the temperature rises to 4000K, then it must add 60 MJ for each kilogram of carbon to be vaporized.

 

We mentioned active cooling. Being able to remove 11 MW/m^2 using a coolant looping through the armor material significantly raises the damage threshold. In this case, it is increased to 19.5 and 25.5 MW/m^2 respectively.

 

Two other techniques covered in Lasers, Mirrors and Star Pyramids are valid in all situations: sloping and rotating.

We can add a good compound slope to the carbon surface: 80 degrees vertical and 67.5 degrees horizontal. This spreads the laser beam over a surface area about 15 times greater. Consequently, the laser now needs to reach an intensity of at least 292.5 MW/m^2.

 

Rotation spreads the beam further. We could have a situation where the beam diameter is 12.56 times smaller than the armor’s circumference, and the average beam intensity is reduced by the same factor.

 

All in all, carbon materials can survive 3.67 GW/m^2 for long periods of time or 4.8 GW/m^2 while ablating.

 

If we have a 100 MW laser with a wavelength of 450nm, being focused by a 4 meter wide mirror, we would be able to damage a simple layer of carbon at a distance of 27,025 km. Using all the techniques just mentioned, the range is shortened to 1,300 km.

 

Why shorten ranges?

 

At very long ranges, space battles are boring with little room for tactical decisions.

From here.

We cover this issue in The Laser Problem: any moderately powerful laser can render maneuvers pointless. Warships cannot escape beam weapons. Even at distances where light lag becomes significant, the beam can be divided to cover a wider area while maintaining its destructive intensity.

 

Shorter ranges allow for acceleration to matter more. Other weapon systems that do not have the supreme range of lasers and particle beams can come into play, such as missiles and kinetics. Angular separation starts to matter, allowing for flanking attacks against warships forced to be more well-rounded to fend off multiple weapons from different directions.

 

It is also important from a narrative and visual perspective. Actions taken have a more immediate effect. Events happen quicker and the danger or relief is greater. It is easier to depict battles and the faster pace will be more agreeable to viewers.

 

Actively cooled armor

 

The 1,300 km figure given in the previous example is situational.

 

It relies on the beam coming in from an ideal angle, which is straight down the edges of the star pyramid shape. If instead it came from a flanking angle, perhaps 80 degrees from the front, then the vertical slope of 80 degrees is completely negated. The benefit from sloping falls from a factor 15 to just 2.6, which makes the laser effective from a distance (15/2.6)^0.5: 2.4 times greater.

 

In other words, a warship that sustain laser fire from the front at 1,300 km is vulnerable to flanking attacks out to 3,122 km.

 

The benefit from rotation also varies with distance, as it changes the size of the beam’s spot relative to the target’s circumference.

 

• Intensity reduction by rotation = Beam spot radius / (3.142 * Armor radius)

The reduction is a dimensionless number.
Beam spot radius and armor radius are in meters.

The beam spot radius decreases linearly as the distance decreases. 

If the warship is being shot at a distance two times closer, then the benefit from rotation is twice as great. This inverse relationship means significant gains from rotation at close distances and small gains at long distances.

 

Distances, mirror sizes, beam wavelength, warship radii and even spot shapes are all variable, so we are unlikely to ever have one single number to describe the effect of rotation.

 

What technique instead can we rely upon in all situations? Can we increase armor effectiveness without the enemy having to sit in certain positions and relative angles?

 

Active cooling is the solution.

We looked at figures of 11 MW/m^2 being removed by flowing water over hot surfaces. Further research shows fusion diverters and rocket engine chambers surviving 100 MW/m^2.

 

We can go further, to absorb much more power.

 

For the sake of comparison, we will be using a standardized model where a flat plate of armor sits under the glare of a laser beam. It conducts heat through a heat exchanging surface to a coolant flow underneath. The armor is 1cm thick and the coolant flow channel 1cm wide. We will calculate the power required to pump coolant to a certain velocity, and won’t allow velocities that cause turbulence in liquid or supersonic compression in gases. The heat is radiated away using simple 1mm thick carbon fiber radiators.

 

There are fourth factors to consider for a proper comparison. The first is the heat that can be removed per square meter. The second is the pumping power requirement. The third is battle damage resistance, which is not as straightforward. The last is the radiator surface area required to handle the heat.

 

We won’t be working out the effect of thermal conductivity as in most cases it is not the limiting factor.

 

Gas cooled armor

Gas cooled rocket nozzle

Gases are an interesting coolant as they have no upper temperature limit. If the armor material is carbon and it can withstand a 3500K temperature, then we can select any gas and heat it up to that temperature in the heat exchanger.

We are looking for the gas with the highest heat capacity. High heat capacity means less gas needs to flow through the heat exchanger to pick up the heat and carry it away. Less gas means reduced pumping power. We also want a gas that doesn’t condense at lower temperatures.

 

Hydrogen is the best. However, it is reactive. It will chemically reduce the carbon and degrade it. Helium is a second-best alternative that is chemically inert. We want it entering the armor at 500K, heating up to 3500K and exiting to be cooled in a radiator back down to 500K. If it does not reach the desired temperature in one pass through the heat exchanger, it can be recirculated through at the cost of doubling the pumping power.

 

The speed of sound in helium at 500K is 1315m/s. At 3500K, it is 3480m/s. We are limited to pumping below the smaller figure, so let’s give it Mach 0.9 (1183m/). This means we can push 1.13 kg/s under each square meter of armor at 1 bar of pressure, and 28.25 kg/s at 25 bar.

 

Helium absorbs 5.19 kJ/kg/K. Over a temperature rise of 500 to 3500K, 28.25 kg/s will absorb 439.7 MW/m^2.

 

Pumping power requirement depends on the pressure drop across the heat exchanger. It will be the highest of all designs considered in this post.

 

The radiators happily handle the heat using 4994 m^2 and 5 tons of mass.

Pumps will most likely resemble those of rocket engines.

Gaseous cooling has the advantage that it can increase its performance with increased pumping pressure, and can maintain some degree of functionality while sustaining battle damage. A sudden increase in temperature is not so dangerous as pressure build-up can be vented into space.

 

However, they have the highest pumping power requirements. While we are ignoring thermal conductivity in our calculations, gaseous cooling using helium has the lowest thermal conductivity of all the fluids we will be considering. This imposes certain design restrictions on the heat exchanger interface with the gas, which is trouble when we want the gas to be flowing through it at near Mach speeds.  

 

Metal vapor cooled armor

 

Helium has high heat capacity but low density. We need a lot of pumping power to push enough volume through the heat exchanger to draw a decent amount of heat away.

Metal vapour prepared for use as a lasing medium.

The gases with the highest densities are metal vapours. The same volume brings a lot more mass throughput and therefore cooling capacity.

 

We want a metal that is dense but boils easily. Mercury is ideal. It boils at 630K, so we’ll set the minimum temperature to 750K to prevent it condensing back into a liquid. As before, we heat it up to 3500K. 

Boiling mercury.

The average density of a mercury vapor at 25 bar, between 750 and 3500K, is 48.7 kg/m^3. It would have a heat capacity of 104 J/kg/K and a speed of sound of 227 m/s at 750K. Putting all this together, we expect to push 99.5 kg/s through the 1cm wide heat exchanger and extract 28.5 MW/m^2. This is a much lower performance than with.

 

Pumping requirements are a significantly lower. It is estimated that mercury vapours remove 5 kW of heat from the armor for every watt of pumping power, which is about 20% better than for helium. Only 1563 m^2 of radiators weighing 1.6 tons are needed to handle the heat.

 

The reduced pumping requirements means that you can use many smaller pumps to push the mercury gas through heat exchangers, which helps with redundancy. However, a sudden pressure drop from holes created in the armor are likely to cause the mercury to suddenly expand, cool and revert to its liquid form. The droplets would quickly block gas flow through narrow channels in the heat exchanger.

 

Another difficulty is that mercury can solidify completely behind unheated or damaged armor. Re-establishing a coolant flow is impossible unless the mercury is boiled again to clear the heat exchanger’s channels. Armor would have to ‘go hot’ before battles and prevented from cooling down too much when not under beam attack.

 

Water cooled armor

 

The traditional cooling liquid is water. It is much denser than a gas, has good thermal conductivity and very high heat capacity.

 

Water’s temperature range is its main limit. If we use it as a liquid, we impose that we use very low temperatures to reject heat from the radiators. Consequently, huge radiating areas would be needed. If we use it as a gas at the same temperatures as helium or metal vapours, it will corrode the heat exchanger and chemically attack everything it touches.

 

Instead, we use a phase change design. High pressures allow water to stay liquid beyond the standard 373K boiling point. It is then heated into steam, up to the maximum temperature the heat exchanger can handle without corrosion. After passing through the radiators, it condenses back into water.

The complicated phase diagram of water.

At 25 bar, we can retain liquid water at 480K. That will be our minimum temperature. It has a density of 1197 kg/m^3. We find steam turbine coatings such as chromium steel able to resist 873K steam for thousands of hours, or chromium-niobium alloys at 923K. That will be our maximum temperature. Steam has an average heat capacity of 2.56 kJ/kg/K between 480 and 920K.

 

The phase change from liquid to gas also absorbs energy. For water, this is a whopping 1840 kJ/kg when starting from 480K.

 

Adding the heat absorbed by the phase change and then rise in temperature, we obtain a total of 2966 kJ/kg.

 

We cannot allow turbulent flow through the heat exchanger, as this drastically decreases the heat transfer rate. Based on a fluid’s Reynolds number and viscosity, we can estimate the maximum velocity before the start of turbulent flow. In this case, it can only be 24 m/s when passing through a 1cm gap.

 

With that flow rate, we get as much as 286.8 kg/s passing through the heat exchanger removing 850 MW/m^2.

 

This is impressive performance. The pumping power requirements are drastically lower than any gas (on the order of 2 kW of heat removed per watt of pumping power). Water has the advantage of gaining most of its heat removing capacity through its phase change when cooling armor from as low as 373K. Increasing the temperature of the armor and therefore of the heat exchanger only improves performance.

Another advantage is that it is likely to serve a second role as propellant on spacecraft. Electric, nuclear and solar rockets can all use water. The consequences are that the coolant needed for the armor’s active cooling does not have to be dead weight, and that after being heated into steam, it can be pushed through a nozzle instead of passed into radiators. During battle, if radiators are hidden or destroyed, the armor can still be kept cool by using water as an open-cycle coolant.   

 

There are downsides though. Water increases in volume over a thousand-fold between liquid and gaseous states. Designing a phase change heat exchanger where liquid enters one side and gas exits the other is tricky to do, and is unlikely to work after receiving battle damage. In fact, creating a hole in the heat exchanger would release pressure and allow the water entering as a liquid to suddenly boil and practically explode in the pipes.

Just like mercury, water can freeze into ice when not heated. Damaged pipes can see themselves blocked by this ice, cutting further cooling. Thankfully, the phase change from liquid into solid also takes a lot of energy, so there is usually plenty of time to re-heat the water and get it flowing again. 

 

The biggest disadvantage is the maximum temperature restriction on the armor and heat exchanger. Above 920K, the thin layer of water or steam in contact with the heat exchanger starts corroding the protective layer quickly. 

If the armor is at 3000K, it will be superheating a small quantity of steam to a vigorous oxygen-hydrogen plasma, even if the average temperatures within the heat exchanger as within bounds. Therefore, if the laser intensity overwhelms the cooling capacity and the armor starts heating up to higher temperatures, we will start to see degraded heat exchangers and a decrease in cooling capacity. This is a self-reinforcing cycle that destroys the armor.  

 

Eutectic cooled armor

Sodium-Potassium coolants for use in TOPAZ-II space reactor.

Liquid coolants have much reduced pumping power requirements. Instead of water with its restrictive temperature limitations, we might select a coolant that can handle much temperatures. 

NaK properties.

Eutetics are mixtures of two or more elements that have a lower melting point than either pure element. A prime example is sodium and potassium used as ‘molten salt’ coolant in nuclear reactors or solar energy storage facilities. Sodium and potassium melt at 371K and 337K respectively, but their eutectic mixture melts at just 260K.

Looks a lot like mercury, but not toxic. From here.

We will be using Galinstan. It is a mixture of gallium, indium and tin. It melts at 254K and boils at 1573K. With a density of 6440 kg/m^3, a heat capacity of 296 J/kg/K and a laminar flow velocity of 85m/s, we find that we can remove 1738MW/m^2 while radiating away heat at 500K.  

 

This incredibly performance is possible due to the fluid’s high density and high viscosity. Pumping requirements will be significant, and you’d need 490,606 m^2 or 491 tons of radiators for every square meter of armor receiving this intensity, but it is worthwhile when it can reduce ranges by so much.

Galinstan would work perfectly inside a liquid droplet radiator.

A note on the radiator requirements: this number is not to be used simply as it is presented. 490,606m^2 are only needed if the enemy beam covers an entire square meter with 1738 MW of power. It is much more likely that a much less powerful beam, for example 100 MW, is focusing its power onto a small spot, perhaps 27cm wide. This gives the same intensity (1738 MW/m^2) but the total heat that must be handled is only 100 MW. The radiator area needed to handle the heat is just 28,218 m^2.

 

One advantage of Galinstan is that it remains liquid at very low temperatures, so there is a much reduced risk of it solidifying and blocking cooling channels. Another is that as an electrically conductive mix of metals, we can use electromagnetic pumping that can end up being more efficient and more damage resistant than conventional pumps.

 

The main disadvantage is that lasers or particle beams can strike multiple spots along an armor surface without warning, so the coolant flow much be able to compensate for any heating across its entire surface. In other words, the pumps must consume large amounts of power to keep Galinstan flowing across the entire armor surface!

 

Another challenge is when beam intensity overwhelms cooling capacity. 1573K is a decently high boiling point, but it is still lower than the 3500K that carbon materials can handle. A hot spot can create vapor bubbles in the Galinstan flow that could cause destructive cavitation or blocked flow in small channels of a hat exchanger.

 

Liquid metal cooled armor

There are liquids that can handle much higher temperatures without boiling. Liquid metals have the highest boiling points.

 

To be a good coolant, we could use a metal with a fairly low melting point, a very high boiling point and the best heat capacity possible. There are many good choices, including plutonium, but we will look at these four in particular: tin, indium, aluminium and cerium.
 

 

Indium has a melting point of 430K and a boiling point of 2345K. It has a density of 7020 kg/m^3 and a heat capacity of 233 J/kg/K. We work out that 2944 MW/m^2 can be removed between 500K and 2300K.

 

Tin has a melting point of 505K and a boiling point of 2875K. It has a density of 6990 kg/m^3 and a heat capacity of 228 J/kg/K. We work out that 3666MW/m^2 can be removed between 550 and 2850K.

 

Aluminium has a melting point of 934K and a boiling point of 2792K. It has a density of 2375 kg/m^3 and a heat capacity of 896 J/kg/K. We work out that 3767 MW/m^2 can be removed between 980 and 2750K.

 

Cerium has a melting point of 1071K and a boiling point of 3697K. It has a density of 6550 kg/m^3 and a heat capacity of 192.4 J/kg/K. We work out that 2999 MW/m^2 can be removed between 1120 and 3500K.

 

For all of the calculations, we limited the flow velocity to 100m/s, despite maximum laminar flow velocities reaching double and more. This gives a more plausible 1m^3 per second volumetric flow rate. 

We have ceramic pumps that can handle liquid metals, and we go further.

The performance of liquid metal cooled armor far exceeds that of other cooling solutions.

 

Indium is at the lowest risk of solidifying in the pipes, but has the highest pumping requirement and imposes the lowest temperature limit on the armor.

Aluminium provides the best performance and the lowest pumping power requirement, but it is the most reactive of the metals and so needs a protective layer in between it and the carbon armor.

 

Cerium, with its very high boiling point, is unlikely to ever create vapour bubbles in the heat exchanger and has the smallest radiators, but it is also at the greatest risk of solidifying inadvertently.

 

Tin is the overall best choice.

 

The danger of course is that a battle starts with the tin in its solid state. Directed energy weapons could add heat to an armor plate too quickly for the tin to melt and start flowing to draw it away. Ideally, the tin is constantly flowing at the minimum temperature, which is 550K in this case. For efficiency’s sake, it could be kept molten using waste heat from a nuclear reactor. However, pumping the metal would consume electrical power that has to be taken away from other systems.

Liquid tin as a thermal transport coolant to a solar energy storage system.

Spaceship designers could make use of the armor layer as another radiator. It would be durable and usable even in battle. Other than the sections under laser attack, it could be rejecting up to 3.5 kW of waste heat for each square meter sitting on liquid tin. This feature could compensate for the extra heat from a nuclear reactor that needs to operate at a higher power level to feed the pumps with electricity.

 

Reduced beam ranges

 

Let’s work out the effective range of a beam weapon facing a carbon armor layer using all the tricks available to us: rotation, compound sloping and active cooling.

As before, we set the weapon to be a 100 MW laser of 450nm wavelength, being focused by a 4m wide mirror.

 

The target will an eight-pointed star (octagram) pyramid 6m wide at the base and 17.3 meters long. This gives it a vertical slope of 80 degrees and a horizontal slope of 67.5 degrees. Each face of the octagram is 1.76m long, which means a total circumference of 28.1m.

 

Before combat, the armor maintains a flow of liquid tin through it. It serves as a radiator which handles about 1.26 MW of heat on its own.

 

During combat, 3.6 GW/m^2 can be absorbed when the liquid tin gets really hot. The maximum temperature we’ll allow is 2800K to prevent any hotspots from boiling the tin. The heated carbon also radiates another 3.5MW/m^2, but this is a tiny contribution.

 

An enemy attacking the pointy end of the star pyramid would face a compound slope that spreads their beam over a surface area 15 times greater. This is a 15-fold reduction in intensity. Working it out, the armor can handle 54.2 GW/m^2 with ideal sloping. If the enemy attacked from the side, they would face only the horizontal slope that reduces intensity by 2.6-fold. The armor can only handle 9.4 GW/m^2 in that case.

 

We can see already that the intensities the armor can handle are much greater than in previous examples.

 

The laser we are considering can only produce an intensity of 9.4 GW/m^2 at distance of 812.6 km. If our warship is outnumbered, it would want to stay at least this distance away from its closest opponent.

 

54.2 GW/m^2 is only possible at a distance of 338.4 km. Our warship can get this close if it is confident that it can always face its pointy end to the enemy.

What about rotation?

 

In these scenarios, the benefits could be massive. At 338.4km, the spot radius of the laser is 2.4cm, and at 812.6km, it is 5.8cm. These are 1170.8x and 484.5x smaller than the circumference of our armor. In ideal conditions, it is a reduction in intensity of 1170.8x and 484.5x. A supremely confident commander could bring our warship to single-digit kilometres in front of the beam and expect to spread its power enough to never overwhelm its cooling capacity!

 

There are important consequences for this sort of cooling capacity and reduction beam ranges. Many depend on the specifics of the setting where space warfare takes place.

Laser attack, captured and edited from this Children of a Dead Earth video.

In general though, we can reasonably expect that battles will revolve around achieving a flanking position, suppressing the cooling capability of the armor or preventing electrical generation that powers the pumps. The latter two objectives can be achieved by pulsed lasers, penetrating particle beams, kinetic strikes and other weaponry that are not continuous beams.

 

At the very least, the threat of giant laser-equipped warships and their dampening effect on any sort of eventful space combat can be reduced or eliminated in science fiction.

[DDE]

@MatterBeam is a SpaceX employee confirmed.

[magnemoe]

Interesting as flanking would work as crossing the T during the age of sails. If you back then managed to get an broadside down the front or back of an enemy your balls would bounce down the entire gun and main deck of the enemy doing way more damage. 
Same is true for all tank combat, take enemy from any angle than frontal. Tank destroyer tanks was typically set up to be very strong in front but weak in other directions as they was designed to fight from positions. 

One fun toy is the laser mirror, drop something stealthy at pretty low speeds this work, I hit it with an high power laser beam who is redirected on you flank, you also have bomb pulsed lasers. 

[sh1pman]

In case of liquid metal cooling, let’s say tin, what about the expansion and contraction of metal during melting and solidification? If you freeze a full bottle of water, the ice takes more volume than liquid water, so the bottle may explode. Are densities of molten and solid tin significantly different? If tin is allowed to solidify when not in battle, this effect can potentially damage or destroy the channels.

[wumpus]

From what I understand, multiple layers of very thin films a long way from the ship are fairly effective, although mostly researched as "navigation shields" in Star Trek lingo (shields to protect you from the dangers of flying through space, not withstanding enemy attack).  They are great for high speed physical strikes (such as nano-asteroids) and may function as "dune shields" limiting torpedoes to speeds that won't vaporize them on contact.

15 hours ago, magnemoe said:

Interesting as flanking would work as crossing the T during the age of sails. If you back then managed to get an broadside down the front or back of an enemy your balls would bounce down the entire gun and main deck of the enemy doing way more damage. 
Same is true for all tank combat, take enemy from any angle than frontal. Tank destroyer tanks was typically set up to be very strong in front but weak in other directions as they was designed to fight from positions. 

One fun toy is the laser mirror, drop something stealthy at pretty low speeds this work, I hit it with an high power laser beam who is redirected on you flank, you also have bomb pulsed lasers. 

Pretty much any space warfare centered around sustained beam fire means that maintaining more than 1 flanking attack (because using three dimensions allows you to angle the surfaces of your heatsink perpendicular to attackers from two directions) means victory.

While the ISS's radiators might be significantly smaller than the solar arrays (and of course it is typically shown to see the solar arrays, so you don't often see the radiators that have to be perpendicular to the solar panels), they do take up a lot of room.  And even in "E.E. Doc Smith" type space opera (think Star Wars or Star Trek, only limited to real physics*) those radiators have to be within orders of magnitude of size *per* *Watt* of the ISS.  So if you are blasting spaceship-destroying lasers willy-nilly, you are going to need either open cooling (with similar issues to the rocket equation as you keep pumping out coolant) or closed cooling with all radiator issues (and vulnerabilities) that implies.

My previous calculations (probably inspired and posted by some other Matterbeam post) included below.

Spoiler

 The theoretical limits are pretty easy to calculate.  The maximum possible efficiency of your power generator is limited by the Carnot cycle limit (and this actually limits real power plants, and even some research car engines are approaching it).

Efficiency<1-(Tc/Th) where Th is the heat of your steam (or whatever) going into a turbine (presumably a combined cycle).  Assuming using Tungsten-Halfnium-unobtanium alloys throughout, this could be somewhere between 3000-4000K.

Tc= the temperature of your heatsink (in Kelvin).  Note that for ISS, this is less than 200K (you could presumably generate power for the ISS at 93% efficiency).

Blackbody radation increases at the fourth power of temperature.   So if you want to produce massive increases of power, you could presumably have the same size radiators and replace the ammonia with molten iron of something and have an efficiency of 10% and have the radiators at 3600K and the steam at 4000K (ok, this assumes your generators and lasers are 100% efficient, but still).  This gives you a 324 fold increase in the amount of power you can produce with the same sized radiators.  I optimized this a little more and came to the conclusion that Th/Tc should equal 3/2 and that for a Th of 4000 you would get 500 times the power over ISS using the same sized radiators (with an efficiency around 33%, but radiating a bit more than a third as much).  The ISS radiators are about 1/10th the size of the solar panels, and not visible from any view directly facing the solar panels (what most pictures show), I'll use this for a "heat sink area unit" for theoretical spacecraft.

REALITY CHECK: This assumes theoretically ideal turbines crafted out of unobtanium (to withstand 4000C), impossibly perfect lasers, and equally impossibly perfect generators (it doesn't spare any heating back from the lasers, and I doubt that such an equation will fit on the envelope used for the above).  And you still have to deal with radiators only a couple orders of magnitude smaller (per Watt) better than a space station lifted into orbit in the 20th century.  So for an *absolutely perfect* system (optimzed to reduce the heatsink), you get ~60MW per ISS-sized-heatsink.  These are fundamental physical limitations and the only ways around them are open cooling cycles, matter that remains solid at 4000K, or perpetual motion machine.  No amount of other tech will change this.  There's a reason I want to fire on the heat sinks.

Note this limitation is unlikely to be an issue outside of deliberate attack.  It places no bounds on the radiators aside from shear size.  You could presumably use gold leaf for the majority of you heatsink (not gold, you need something that won't melt at 2600K).  It is only when you need something that can endure deliberate attack does the issue of raw surface area become an issue.  Note that I suspect that with tech in barely doable (in parts, not all at once) 21st century tech we could bump this up to 30MW per ISS heatsink, but after that the asymptote gets *steep*.  I'm guessing that any "missile based" laser will fire only briefly and use some sort of open cooling scheme.  Open loop cooling works great if you need high-Isp "flanking speed', but also has the same tyranny as the rocket equation (bring *lots* of mass).  Liquid hydrogen gives you great Isp, but little cooling.  I suspect things like tungsten, halfnium, and depleted uranium (or just spent fuel rods) would be good for open loop cooling.

* E.E. "Doc" Smith's physics might be seem a little iffy, but seeing how he wrote his famous works in the 1920s-1930s I'll cut him that slack.

Edited May 13, 2019 by wumpus
added forgotten footnote

[KerikBalm]

This all deals with lasers that melt through hulls, but if the laser output is high enough, as in a pulsed laser (keeping in mind, we've already meta petawatt lasers... but their total energy output is rather modest because the duration is so short).

In this case, the surface of the target material is vaporized, but vaporized so fast that there is an explosive shockwave. You may only vaporize 10% of the thickness of the armor, but the showave fractures the armor and damages the insides of the ship.

Active cooling would be useless against pulse lasers designed to affect the target like that.

For a continuous laser trying to melt through a target, sure it works. Most lasers fall below 50% efficiency, so the firing ship will have to have an active cooling system that is signficantly more powerful than the target's, it would become a heat rejection contest (which has come up in sci fi scenarios before, or a related heat sink contest, where deployed radiators is like striking a flar and signalling surrender).

As long as the laser is below 50% efficiency, the only hope of working is that the energy is dispersed in the firing ship, and concentrated in the fired upon ship. The more and more concentrated the energy is, the less active cooling will be able to take away the heat fast enough.

So the best scenario for a laser would be petawatt outputs, with gigajoule pulses, of very focused hard xray lasers, fired from large focusing arrays to improve the ratio of energy concentration at the target vs the generator

http://www.projectrho.com/public_html/rocket/sidearmenergy.php#blaster

 

[KerikBalm]

Just to fill in some numbers, from here:

http://panoptesv.com/SciFi/LaserDeathRay/HardX.html

I'm taking a rather short wavelength, but not the limit, of 6 e-11 meters... 3/4 of the way from the end of "soft X-rays" to the start of "gamma rays"... still well within what is considered Hard Xrays, which are pretty much the limit as to what would be possible to focus.

Then for power, I'm assuming 1 petawatt, the most powerful we've built is 1.3 PW:

https://en.wikipedia.org/wiki/Laser#Examples_by_power

Then I'm going to assume a 20 meter radius lens because.. well why not, we're talking fantasticly powerful future spacewarships.

I'm going to assume a beam duration of 0.1 microseconds/100 nanoseconds... beam durations can be in the femtoseconds, or much longer, this seems reasonable to me. The lenght doesn't really matter for this calculator and what I'm getting at.

Duty cycle: 1, I'm not going to assume its firing the whole time, not cycling on/off.

This is a Hard X-ray laser, firing a 100 Mega-Joule pulse, at a 2 Peta-Watt energy output. 100 Mega joules isn't soo much, the P-51's engine had over a 1 MW output, so thats the energy that engine put out over 100 seconds.

 

Now to the online calculator:

https://web.archive.org/web/20170412014001/http://www.5596.org/cgi-bin/laser.php

Laser parameters entered, select carbon as the armor (graphite, it doesn't really matter here though, its an obscenely powerful laser).

Then for ranges (in meters), I'm going to use: 0.15e+11, 0.375e+11, 0.75e+11, 1.5e+11, 3e+11, 4.5e+11 : in other words; 0.1 AU, 1/4 AU, 1/2 AU, 1 AU, 2 AU, 3 AU, 4 AU

Click fire lasers!

Notice that it says impulse shock for all of the distances!

Quote

Impulse shock indicates that the armor is vaporized at a rate exceeding the speed of sound, tearing and damaging the surrounding hull. You can pretty much consider the compartment utterly destroyed

Change the last one from 4 AU to 5 AU, and we no longer get impulse shock. You may look at the depth of target material vaporized, and think its quite small...

But that is how much was vaporized, not how much was blasted away.

http://panoptesv.com/SciFi/LaserDeathRay/DamageFromLaser.php

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The beam is initially incident on the target. A thin layer of target material is flashed to plasma, and the plasma absorbs the rest of the pulse. At this point, the plasma is at extremely high pressure, much greater pressure than the strength of the target material.

The expanding plasma launches a blast wave. A shell of hot vapor blasts out into the air, while a shell of super-hot, highly compressed material propagates into the target. Pressures are so high the target material flows like a fluid.

As the blast front propagates into the material, it slows down.

Eventually, the blast wave has done so much work pushing through the target material and has spread out so much that the pressure is no longer far in excess of the material strength. The flowing blast wave, no longer able to bull its way through the target material, is instead re-directed along the crater walls and splashes out into the air.

Once the fluidized material that made up the blast wave has squirted out of the crater along the crater sides and into the air, what is left is a permanent crater, surrounded by a layer of compressed and deformed material, and beyond that the mostly undamaged target material.

Get your peak output high enough, and you're blowing away material with high pressure, not melting or vaporizing your way through it.

With a building sized laser (such as on a capital ship), you can cause explosions on the surface of enemy vessels multiple AU away.

Keep in mind, effects like this at 3 AU, means that the laser is taking about 25 minutes to reach the target...

Put one of these on an asteroid, and it can have a massive heat sink.... make it a bit bigger to get a 4 AU range, park an asteroid at Jupiters L1 point, and put 2 other's at L4 and L5, and all over the inner solar system is within range...

Actively cooled armor wouldn't help in such a scenario. The laser installations would have a larger heat sink and heat rejection capacity, and they'could overcome the heat rejection capacity of a ship even if the laser was operating in continuous rather than pulsed mode.

Pulsed lasers render active cooling pointless.

*edit*, used the other calculator; here are the results:

Spoiler

Beam parameters
Beam power: 1000 TW  (1 PW)
Beam diameter at target: 27.5 cm (what the other calculator said radius would be at 1 AU, assuming a wavelength of 6e-11, and 20m radius on the focuser)
Beam duration: 0.1 μs
Beam energy: 100 MJ
Number of pulses: 10
Time between pulses: 0.5 μs
Total Beam energy: 1000 MJ (1 GigaJoule)
Total Beam duration: 5 μs

Environmental parameters
Ambient temperature: 300 K
Ambient pressure: 0.101 Pa

Material properties (selected Graphite, cutting them out here for brevity)

Drilling conditions
Vapor pressure: 998 GPa
Surface temperature: 686300 K
Melt thickness: 1.81E-9 m
Vapor Jet Speed: 23 km/s

Material damage
Blasting Speed: 30 km/s
Width of hole: 29.8 cm
Depth of hole: 1.49 cm
Time to excavate hole: 0.497 μs
Aspect ratio: 0.05

Material damage from pulse train
Width of hole: 33.5 cm
Depth of hole: 29.9 cm
Aspect ratio: 0.892

So... a series of 1 PW pulses totaling 1 GJ of energy would penetrate 30 centimeters into graphite armor at 0.5 AU from such a laser.

Edited May 15, 2019 by KerikBalm